Homework Helper Chapter 1

Transcription

Topic 1
TOPIC OVERVIEW
VOCABULARY
1-1
Points, Lines, and Planes
1-2
Measuring Segments
1-3
Measuring Angles
1-4
Exploring Angle Pairs
1-5
Basic Constructions
DIGITAL
Tools of Geometry
APPS
English/Spanish Vocabulary Audio Online:
EnglishSpanish
angle bisector, p. 22
bisectriz de un ángulo
collinear points, p. 4
puntos colineales
congruent segments, p. 10
segmentos congruentes
construction, p. 27construcción
linear pair, p. 22
par lineal
measure of an angle, p. 16
medida de un ángulo
perpendicular bisector, p. 27mediatriz
postulate, p. 4postulado
point, p. 4punto
ray, p. 5semirrecta
segment bisector, p. 11
bisectriz de un segmento
vertical angles, p. 22
ángulos opuestos por el vértice
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2
Topic 1 Tools of Geometry
3--Act Math
The Mystery
Spokes
Some photos are taken in
such a way that it’s difficult to
determine exactly what the
picture shows. Sometimes it’s
because the photo is a close-up
of part of an object, and you
do not see the entire object.
Other times, it might be because
the photographer used special
effects when taking the photo.
You can often use clues from
the photo to determine what is
in the photo and also what the
rest of the object might look like.
What clues would you look for?
Think about this as you watch
this 3-Act Math video.
Scan page to see a video
for this 3-Act Math Task.
If You Need Help . . .
Vocabulary Online
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terms in both English and
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3
1-1 Points, Lines, and Planes
TEKS FOCUS
VOCABULARY
•Axiom – See postulate.
•Collinear points – Collinear points lie on
TEKS (4)(A) Distinguish
between undefined terms,
definitions, postulates,
conjectures, and theorems.
the same line.
•Coplanar – Coplanar points and lines lie in
TEKS (1)(D) Communicate
mathematical ideas,
reasoning, and their
implications using multiple
representations, including
symbols, diagrams, graphs,
and language as appropriate.
the same plane.
•Intersection – The intersection of two or
more geometric figures is the set of points
the figures have in common.
•Line – A line is represented by a straight
path that extends in two opposite directions
without end and has no thickness. A line
contains infinitely many points.
Additional TEKS (1)(F)
•Opposite rays – Opposite rays are two
rays that share the same endpoint and
form a line.
•Plane – A plane is represented by a flat
surface that extends without end and has
no thickness. A plane contains infinitely
many lines.
•Point – A point indicates a location and
has no size.
•Postulate – A postulate, or axiom, is an
accepted statement of fact.
•Ray – A ray is part of a line that consists of
one endpoint and all points of the line on
one side of the endpoint.
•Segment – A segment is part of a line that
consists of two endpoints and all points
between them.
•Space – Space is the set of all points in
three dimensions.
•Implication – a conclusion that follows
from previously stated ideas or reasoning
without being explicitly stated
•Representation – a way to display or
describe information. You can use a
representation to present mathematical
ideas and data.
ESSENTIAL UNDERSTANDING
Geometry is a mathematical system built on accepted facts, basic terms, and definitions.
4
Key Concept Undefined Terms
Term Description
How to Name It
A point indicates a location and has
no size.
You can represent a point by a dot and
name it by a capital letter, such as A.
A line is represented by a straight path
that extends in two opposite directions
without end and has no thickness. A
line contains infinitely many points.
You can name a line by any< two
>
points on the<line,
> such as AB (read
“line AB”) or BA , or by a single
lowercase letter, such as line /.
A plane is represented by a flat
surface that extends without end
and has no thickness. A plane
contains infinitely many lines.
You can name a plane by a capital
letter, such as plane P, or by at least
three points in the plane that do not all
lie on the same line, such as plane ABC.
Diagram
A
B
A hsm11gmse_0102_t00699.ai
A
B
P
C
hsm11gmse_0102_t00700.ai
Lesson 1-1 Points, Lines, and Planes
Key Concept Defined Terms
Definition
How to Name It
Diagram
A segment is part of a line that
consists of two endpoints and all
points between them.
You can name a segment by
its two endpoints, such as AB
(read “segment AB”) or BA.
A ray is part of a line that consists
of one endpoint and all the points
of the line on one side of the
endpoint.
You can name a ray by its endpoint
and another
point on the ray, such
>
as AB (read “ray AB”). The order of
points indicates the ray’s direction.
A
Opposite rays are two rays that
share the same endpoint and form
a line.
You can name opposite rays by their
shared endpoint and any
> other point
>
on each ray, such as CA and CB .
A
C
B
hsm11gmse_0102_t00704
A
B
B
hsm11gmse_0102_t007
hsm11gmse_0102_t00705.a
Postulate 1-1
Through any two points there is exactly one line.
t
A
B
Line t passes through points A and B. Line t is the only line that passes
through both points.
Postulate 1-2
A
If two distinct lines intersect, then they intersect in exactly one point.
< >
< >
AE and DB intersect in point C.
B
C
D
Postulate 1-3
If two distinct planes intersect, then they intersect in exactly one line.
< >
Plane RST and plane WST intersect in ST .
R
T
S
E
W
Postulate 1-4
Through any three noncollinear points there is exactly one plane.
Points Q, R, and S are noncollinear. Plane P is the
only plane that contains them.
R
S
P
Q
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5
Problem 1
Why can figures have
more than one name?
Lines and planes are
made up of many points.
You can choose any
two points on a line
and any three or more
noncollinear points in a
plane for the name.
Naming Points, Lines, and Planes
< >
T
A What are two other ways to name QT ?
<
<
>
R Q
>
Two other ways to name QT are TQ and line m.
B What are two other ways to name plane P?
V
P
N
Two other ways to name plane P are plane RQV and plane RSV.
S m
C What are the names of three collinear points? What are the
names of four coplanar points?
Points R, Q, and S are collinear. Points R, Q, S, and V are coplanar.
Problem 2
TEKS Process Standard (1)(D)
Naming Segments and Rays
How do you make
sure you name all
the rays?
Each point on the line is
an endpoint for a ray. At
each point, follow the
line both left and right
to see if you can find a
second point to name
the ray.
A What are the names of the segments in the figure at the right?
The three segments are DE or ED, EF or FE, and DF or FD.
B What are the names of the rays in the figure?
>
>
>
>
>
D
>
The four rays are DE or DF , ED , EF , and FD or FE .
C Which of the rays in part (B) are opposite rays?
F
E
>
>
The opposite rays are ED and EF .
Problem 3
Distinguishing Between Undefined Terms and Definitions
Which term below is undefined and which term has a definition? Write the
definition of the defined term.
pointray
Point is an undefined term, whereas ray has a definition. A ray is part of a line that
consists of one endpoint and all of the points of the line on one side of the endpoint.
Problem 4
TEKS Process Standard (1)(F)
Finding the Intersection of Two Planes
Each surface of the box at the right represents part of a plane.
What is the intersection of plane ADC and plane BFG?
D
A
E
Plane ADC and
plane BFG
The intersection of
the two planes
H
C
B
G
F
Find the points that the
planes have in common.
continued on next page ▶
6
Problem 4
Is the intersection a
segment?
No. The intersection of
the sides of the box is
a segment, but planes
continue without end.
The intersection is a line.
continued
D
A
C
B
H
E
Focus on plane ADC and
plane BFG to see where
they intersect.
G
F
D
A
C
B
H
E
G
You can see that both
planes contain point B
and point C.
F
< >
The planes intersect in BC .
Problem
5
Using Postulate 1-4
How can you find the
plane?
Try to draw all the lines
that contain two of the
three given points. You
will begin to see a plane
form.
Use the figure at the right.
M
J
A What plane contains points N, P, and Q? Shade the plane.
M
J
R
N
L
K
Q
K
R
N
The plane on the bottom of the figure
contains points N, P, and Q.
L
Q
P
P
B What plane contains points J, M, and Q? Shade the plane.
M
L
J
The plane that passes at a slant through
K
Q
NLINE
HO
ME
RK
O
WO
N
the figure contains points J, M, and Q.
P
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
m
Use the figure at the right for Exercises 1–3.
1.
What are two other ways to name plane C?
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2.
Name three collinear points.
3.
Name four coplanar points.
n
C
E
B
F
G
Use Multiple Representations to Communicate Mathematical
Ideas (1)(D) Use the figure at the right for Exercises 4 and 5.
4.
Name the segments in the figure.
5.
Name the rays in the figure.
R S
T W
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7
Use the figure at the right for Exercises 6–15.
X
U
Name the intersection of each pair of planes.
6.
planes QRS and RSW
W
V
7.planes UXV and WVS
Name two planes that intersect in the given line.
< >
< >
< >
< >
8.
QU 9.
TS 10.
XT 11.
VW
Q
T
S
R
Copy the figure. Shade the plane that contains the given points.
12.R, V, W13.
U, V, W14.
U, X, S15.
T, U, V
Postulate 1-4 states that any three noncollinear points lie in
exactly one plane. Find the plane that contains the first three
points listed. Then determine whether the fourth point is in that
plane. Write coplanar or noncoplanar to describe the points.
16.Z, S, Y, C
Z
V
17.S, V, C, Y
X
S
Y U
C
18.Display Mathematical< Ideas
<Draw
> < (1)(G) >
> a figure with points B, C, D,
E, F, and G that shows CD , BG , and EF , with one of the points on all
three lines.
19.Your friend drew the diagram at the right to prove to you that
two planes can intersect in exactly one point. Describe your
friend’s error.
20.Which term below is undefined and which term has a definition?
Write the definition of the defined term.
line opposite rays
21.Analyze Mathematical Relationships (1)(F) If one ray contains
another ray, are they the same ray? Explain.
For Exercises 22–25, determine whether each statement is always,
sometimes, or never true.
< >
< >
22.TQ and QT are the same line.
>
>
24.JK and JL are the same ray.
23.Two intersecting lines are coplanar.
25.Four points are coplanar.
26.Use the diagram at the right. How many planes contain both the line
and the point?
< >
a.FG and point P
P
Q
< >
b. EP and point G
c.Explain Mathematical Ideas (1)(G) What do you think is true of
a line and a point not on the line? Explain.
G
H
E
F
8
STEM
27.Apply Mathematics (1)(A) A cell phone
tower at point A receives a cell phone signal
from the southeast. A cell phone tower at
point B receives a signal from the same cell
phone from due west. Trace the diagram
at the right and find the location of the cell
phone. Describe how Postulates 1-1 and
1-2 help you locate the phone.
A
SE
28.You can represent the hands on a clock at
6:00 as opposite rays. Estimate the other
11 times on a clock that you can represent as
opposite rays.
B
W
29.Apply Mathematics (1)(A) What are some basic words in English
that are difficult to define?
HSM11GMSE_0102_a00319
2nd pass 01-02-09
Durke
30.a.Explain Mathematical Ideas (1)(G) Suppose two points are in plane P.
Explain why the line containing the points is also in plane P.
b.Use Representations to Communicate Mathematical Ideas (1)(E) Suppose two lines intersect. How many planes do you think contain
both lines? Use the diagram at the right and your answer to part (a) to
explain your answer.
A
B
C
Display Mathematical Ideas (1)(G) Graph the points and state whether
they are collinear.
31.(1, 1), (4, 4), ( -3, -3)
32.(2, 4), (4, 6), (0, 2)
33.(0, 0), ( -5, 1), (6, -2)
Use a Problem-Solving Model (1)(B) Suppose you pick points at
random from A, B, C, and D shown at the right. Find the probability
that the number of points given meets the condition stated.
34.2 points, collinear
35.3 points, collinear
D
A
B
C
TEXAS Test Practice
36.Which geometric term is undefined?
A.segment
B.collinear
C.ray
D.plane
37.You want to cut a block of cheese into four pieces. What is the least number
of cuts you need to make?
F.
2
G.3 H.4
J.5
A
38.The figure at the right is called a tetrahedron.
a.Name all the planes that form the surfaces of the tetrahedron.
b.Name all the lines that intersect at D.
D
B
C
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9
1-2 Measuring Segments
TEKS FOCUS
VOCABULARY
TEKS (2)(A) Determine the coordinates
of a point that is a given fractional
distance less than one from one end of
a line segment to the other in one- and
two-dimensional coordinate systems,
including finding the midpoint.
TEKS (1)(D) Communicate
mathematical ideas, reasoning, and
their implications using multiple
representations, including symbols,
diagrams, graphs, and language as
appropriate.
Additional TEKS (1)(F), (1)(G)
•Congruent segments – Two
•Segment bisector – A segment
segments are congruent (≅)
segments if they have the same
length.
bisector is a point, a line, a
ray, or another segment that
intersects a segment at its
midpoint (or bisects it).
•Coordinate – The coordinate of
a point is the real number that
corresponds to a point.
•Implication – a conclusion that
follows from previously stated
ideas or reasoning without being
explicitly stated
•Distance – The distance between
two points is the absolute value of
the difference of their coordinates.
•Representation – a way to display
•Midpoint – The midpoint of a
or describe information. You can
use a representation to present
mathematical ideas and data.
segment is a point that divides
the segment into two congruent
segments.
You can use number operations to find and compare the lengths of segments.
Postulate 1-5 Ruler Postulate
Every point on a line can be paired with a real number. This
makes a one-to-one correspondence between the points on
the line and the real numbers.
A
B
The real number that corresponds to a point is called the
coordinate of the point.
a
b
coordinate of A
Postulate 1-6 Segment Addition Postulate
If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.
AB
A
BC
B
C
AC
10
Lesson 1-2 Measuring Segments
coordinate of B
Key Concept Midpoint and Segment Bisector
The midpoint of a segment is a point that divides the segment into two congruent
segments. A point, line, ray, or other segment that intersects a segment at its midpoint is
said to bisect the segment. That point, line, ray, or segment is called a segment bisector.
B is the midpoint
of AC.
is a segment
bisector of AC.
A
B
C
Key Concept Midpoint Formula on a Number Line
Description
Formula
Diagram
The coordinate of the midpoint
is the average or mean of the
coordinates of the endpoints.
The coordinate of the
midpoint M of AB is a +2 b .
A
a
Problem 1
B
ab
2
b
Measuring Segment Lengths
What are you trying
to find?
ST represents the length
of ST , so you are trying
to find the distance
between points S and T.
M
What is ST ?
S
6 4 2
0
2
4
6
T
U
V
8
10 12 14 16
The distance between points S and T is the absolute value of the difference of their
coordinates, or |s - t|. This value is also called ST, or the length of ST .
Ruler Postulate
The coordinate
of S is -4.
The coordinate of T is 8.
ST = 0 -4 - 8 0 = 0 -12 0
= 12
Ruler Postulate
Definition of distance
Subtract.
Find the absolute value.
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11
Problem 2
Using the Segment Addition Postulate
Algebra If EG = 59, what are EF and FG?
8x 14
4x 1
E
EG = 59
EF = 8x - 14
FG = 4x + 1
EF and FG
F
G
Use the Segment Addition Postulate to
write an equation.
EF + FG = EG
(8x - 14) + (4x + 1) = 59 12x - 13 = 59 Combine like terms.
12x = 72 Add 13 to each side.
x=6 Segment Addition Postulate
Substitute.
Divide each side by 12.
Use the value of x to find EF and FG.
EF = 8x - 14 = 8(6) - 14 = 48 - 14 = 34
FG = 4x + 1 = 4(6) + 1 = 24 + 1 = 25
Substitute 6 for x.
Problem 3
Comparing Segment Lengths
How do you know
if segments are
congruent?
Congruent segments
have the same length. So
find and compare the
lengths of AC and BD.
Are AC and BD congruent?
A
6 4 2
B
0
2
C
4
D
6
8
AC = 0 -2 - 5 0 = 0 -7 0 = 7
BD = 0 3 - 10 0 = 0 -7 0 = 7
E
10 12 14 16
Definition of distance
Yes. AC = BD, so AC ≅ BD.
Problem 4
Finding the Midpoint
Will the midpoint be
positive or negative?
Since the positive number
has a greater absolute
value, the midpoint of
- 4 and 9 will be positive.
AB has endpoints at −4 and 9. What is the
coordinate of its midpoint?
8 6 4 2
B
0
2
4
6
8 10 12
Let a = - 4 and b = 9.
M=
a + b -4 + 9 5
= 2 = 2.5
2 =
2
The coordinate of the midpoint of AB is 2.5.
12
A
Problem 5
Using the Midpoint
How can you use
algebra to solve the
problem?
The lengths of the
congruent segments
are given as algebraic
expressions. You can set
the expressions equal to
each other.
Algebra Q is the midpoint of PR. What are PQ, QR, and PR?
6x 7
5x 1
P
Q
R
Step 1Find x.
PQ = QR
Definition of midpoint
6x - 7 = 5x +hsm11gmse_0103_t00762.ai
1
Substitute.
x-7=1
Subtract 5x from each side.
x=8
Add 7 to each side.
Step 2Find PQ and QR.
PQ = 6x - 7
QR = 5x + 1
= 6(8) - 7
Substitute 8 for x.
= 5(8) + 1
= 41
Simplify.
= 41
Step 3Find PR.
PR = PQ + QR
= 41 + 41 Substitute.
= 82
Simplify.
Segment Addition Postulate
PQ and QR are both 41. PR is 82.
Problem 6
Determining the Coordinate of a Point on a Line Segment
In AB, point C is 23 the distance from point A to point B.
What is the coordinate of point C?
A
0
To which coordinate
do you add or
subtract?
Use the point you are
starting from, which is
point A.
6
AB = 0 15 - 6 0 = 9
23
C
# 9 = 6
6 + 6 = 12
B
15
Find AB.
Find 23 AB.
2
Add 3 AB to the coordinate of point A.
Add, because the starting coordinate
is less than the ending coordinate.
The coordinate of point C is 12.
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13
HO
ME
RK
O
NLINE
WO
B
C
D
E
8 6
1
3
7
A
Find the length of each segment.
1.
AB
2.BD
Use the figure at the right for Exercises 3 and 4.
3.
If RS = 15 and ST = 9, then RT = ■.
R
4.
If ST = 15 and RT = 40, then RS = ■.
T
S
5.
Apply Mathematics (1)(A) The numbers labeled on the map of Florida are mile
markers. Assume that Route 10 between Quincy and Jacksonville is straight.
Quincy
199
Monticello
181
Tallahassee
225
Madison
10
283
251
Live Oak
Macclenny
303
Jacksonville
10
335
Lake City
95
357
95
Suppose you drive at an average speed of 55 mi/h. How long will it take HSM11GMSE_0103_a00326
to get from Live Oak to Jacksonville?
2nd pass 01-05-09
6.
On a number line, A is at - 2 and B is at 4. What is the coordinate
of C, which is
Durke
2
3 of the way from A to B?
7.
Analyze Mathematical Relationships (1)(F) A is the midpoint of XY .
a.Find XA.
3x
5x 6
X
A
< >
8.
Suppose point E has a coordinate of 3 on EG and EG = 5. What are the possible
coordinates of point G?
Y
b.Find AY and XY.
Use the diagram at the right for Exercises 9 and 10.
A
9.
If AD = 12 and AC = 4y - 36, find the value of y.
Then find AC and DC.
B
hsm11gmse_0103_t00822
D
E
C
10.If ED = x + 4 and DB = 3x - 8, find ED, DB, and EB.
11.Explain Mathematical Ideas (1)(G) Suppose you know PQ and QR.
Can you use the Segment Addition Postulate to find PR? Explain.
4x 3
hsm11gmse_0103_t00825
12.Use Multiple Representations to Communicate
Mathematical Ideas (1)(D) Use the diagram at the right.
2x 3
x
a.What algebraic expression represents GK?
b.If GK = 30, what are GH and JK?
G
H
J
K
Determine the coordinate of the midpoint of the segment
with the given endpoints.
13.2 and 4
14
14. -9 and 6
15.2 and -5
hsm11gmse_0103_t00826
16.C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD,
F is the midpoint of ED, G is the midpoint of EF , and H is the midpoint of DB.
If DC = 16, what is GH?
Connect Mathematical Ideas (1)(F) Use the number line below for Exercises 17–20.
Tell whether the segments are congruent.
L
M
10
N
5
P
0
5
Q
10
15
17.LN and MQ18.
MP and NQ19.
MN and PQ20.
LP and MQ
21.A driver reads the highway sign and says, “It’s 145 miles from
Mitchell
to Watertown.” What error did the driver make? Explain.
Analyze Mathematical Relationships (1)(F) For Exercises 22–24,
3
determine the coordinate of the point that is 5 the distance from the
first point given to the second point given.
22.Point A has coordinate 3; point B has coordinate 5.
23.Point Y has coordinate 11; point X has coordinate -4.
24.Point N has coordinate 12 ; point P has coordinate 0.
25.A bowling alley is 5 miles from your house on a straight road. The supermarket is
on the same road, 13 of the way from your house to the bowling alley. If you draw
a number line and put your house at coordinate 0, what is the coordinate of the
supermarket?
TEXAS Test Practice
26.Which statement is true based on the diagram?
A
B
C
D
E
7
3
2
4
8
A.BC ≅ CE
C. AC + BD = AD
B.BD 6 CD D. AC + CD = AD
hsm11gmse_0103_t00847
27.Points
X, Y, and Z are collinear, and Y is between X and Z. Which statement must
be true?
F.
XY = YZ
G.XZ - XY = YZ H. XY + XZ = YZ
J. XZ = XY - YZ
28.Name four points that are collinear in the figure
at the right.
E
A
B
C
D
F
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hsm11gmse_0103_t00819
15
1-3 Measuring Angles
TEKS FOCUS
VOCABULARY
Foundational To TEKS (6)(A) Verify
theorems about angles formed by the
intersection of lines and line segments,
including vertical angles, and angles
formed by parallel lines cut by a transversal
and prove equidistance between the
endpoints of a segment and points on its
perpendicular bisector and apply these
relationships to solve problems.
•Acute angle – An acute angle is an •Sides of an angle – The sides of an
TEKS (1)(D) Communicate mathematical
ideas, reasoning, and their implications
using multiple representations, including
symbols, diagrams, graphs, and language as
appropriate.
•Measure of an angle – The
Additional TEKS (1)(C)
angle whose measure is between
0 and 90.
•Angle – An angle is formed by two
rays with the same endpoint.
•Congruent angles – Congruent
angles are angles with the same
measure. If m∠A = m∠B, then
∠A ≅ ∠B.
measure of an angle is the
absolute value of the difference of
the real numbers paired with the
sides of the angle.
•Obtuse angle – An obtuse angle
is an angle whose measure is
between 90 and 180.
angle are the rays of the angle.
•Straight angle – A straight angle is
an angle whose measure is 180.
•Vertex of an angle – The vertex
of an angle is the common
endpoint of the two rays that
form the angle.
explicitly stated
•Right angle – A right angle is an
angle whose measure is 90.
You can use number operations to find and compare the measures of angles.
Key Concept Angle
Definition
An angle is formed by two rays
with the same endpoint.
How to Name It
You can name an angle by
The rays are the sides of the
angle. The endpoint is the vertex
of the angle.
• a point on each ray and the
vertex, ∠BAC or ∠CAB
Diagram
B
• its vertex, ∠A
• a number, ∠1
A
1
C
>
The sides
> of the angle are AB
and AC .
The vertex is A.
16
Lesson 1-3 Measuring Angles
Postulate 1-7 Protractor Postulate
>
>
>
Consider OB and a point A on one side of OB . Every ray of the form OA can be
paired one to one with a real number from 0 to 180.
A
0 10
20
180 170 1
60 30
15
0 1 40
40
O
1
inches
170 180
60
0 1 0 10 0
15
2
0
0
14 0 3
4
80 90 100 11
0 1
70
90 80
20
70
60 110 100
60 13
0
0
50 12
50
0
13
2
3
4
5
B
6
Key Concept Types of Angles
Consider the diagram below. The measure of ∠COD
is>the absolute value
of the
>
>
difference of the
real
numbers
paired
with
OC
and
OD
.
That
is,
if
OC
corresponds
>
with c, and OD corresponds with d, then m∠COD = 0 c - d 0 .
0 10
20
180 170 1
60 30
15
0 1 40
40
C
1
inches
acute angle
c
d
2
right angle
O
3
4
170 180
60
0 1 0 10 0
15
2
0
0
14 0 3
4
80 90 100 11
0 1
70
90 80
20
70
60 110 100
60 13
0
0
2
0
1
5
50
0
13
5
obtuse angle
D
6
straight angle
x
x
0 x 90
x
x
x 90
90 x 180
x 180
Postulate 1-8 Angle Addition Postulate
hsm11gsme_0104_t00874.ai
If point B is in the interior of ∠AOC, then m∠AOB + m∠BOC = m∠AOC.
A
B
O
C
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Problem 1
Naming Angles
What
> rays> form j1?
MJ and MK form ∠1.
K
J
What are two other names for j1?
1 2
∠JMK and ∠KMJ are also names for ∠1.
L
M
Problem 2
TEKS Process Standard (1)(C)
Measuring and Classifying Angles
What are the measures of jLKN , jJKL, and jJKN ? Classify each angle
as acute, right, obtuse, or straight.
J
80 90 100 11
0 1
70
90 80
20
70
60 110 100
60 13
0
0
2
0
1
5
50
0
13
Do the classifications
make sense?
Yes. In each case, the
classification agrees
with what you see in the
diagram.
inches
1
2
K
M
170 180
160
0
10 0
15
20
0
0
14 0 3
4
H
0 10
20
180 170 1
60 30
15
0 1 40
40
L
3
4
5
6
N
Use the definition of the measure of an angle to calculate each measure.
m∠LKN = 0 145 - 0 0 = 145; ∠LKN is obtuse.
hsm11gmse_0104_t00875
m∠JKL = 0 90 - 145 0 = 0 -55 0 = 55; ∠JKL is acute.
m∠JKN = 0 90 - 0 0 = 90; ∠JKN is right.
Problem 3
Using Congruent Angles
Look at the diagram.
What do the angle
marks tell you?
The angle marks tell
you which angles are
congruent.
Sports Synchronized swimmers form angles
with their bodies, as shown in the photo.
If mjGHJ = 90, what is mjKLM?
∠GHJ ≅ ∠KLM because they
both have two arcs.
So m∠GHJ = m∠KLM = 90.
D
B
C
F
E
A
M
K
J
G
18
L
H
Problem 4
Using the Angle Addition Postulate
How can you use the
expressions in the
diagram?
The algebraic expressions
represent the measures
of the smaller angles,
so they add up to the
measure of the larger
angle.
Algebra If mjRQT = 155, what are mjRQS and mjTQS?
m∠RQS + m∠TQS = m∠RQT
1 4x - 20 2 + 1 3x + 14 2 = 155
Substitute.
7x - 6 = 155
Combine like terms.
7x = 161
Add 6 to each side.
x = 23
RK
O
HO
ME
WO
m∠TQS = 3x + 14 = 3 1 23 2 + 14 = 69 + 14 = 83
(3x 14)
(4x 20)
m∠RQS = 4x - 20 = 4 1 23 2 - 20 = 92 - 20 = 72
NLINE
S
Angle Addition Postulate
R
T
Q
Substitute 23 for x.
Name each shaded angle in three different ways.
1.
Y
K
3.
C
Z
A
1
2
J
Use the diagram at the right. Find the measure of each
angle. Then classify the angle as acute, right, obtuse,
or straight.
D
E
80 90 100 11
0 1
70
90 80
20
70
60 110 100
60 13
0
0
2
0
1
5
50
0
13
5.
∠DAF
C
6.∠BAE
B
inches
1
2
A
170 180
160
0
10 0
15
20
0
0
14 0 3
4
4.
∠EAF
Draw a figure that fits each description.
L
M
B
0 10
20
180 170 1
60 30
15
0 1 40
40
2.
X
3
4
5
6
F
7.
an obtuse angle, ∠RST
8.
an acute angle, ∠GHJ
Use the diagram at the right. Complete each statement.
D
C
F
9.
∠CBJ ≅ ■
10.If m∠EFD = 75, then m∠JAB = ■.
11.If m∠GHF = 130, then m∠JBC = ■.
J
B
E
G
H
A
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12.Apply Mathematics (1)(A) A pair of earrings has blue wedges that are all the
same size. One earring has a 25° yellow wedge. The other has a 14° yellow wedge.
Find the angle measure of a blue wedge.
25
14
13.Create Representations to Communicate Mathematical Ideas (1)(E) Sketch
a right angle with vertex V. Name it ∠1. Then sketch a 135° angle that shares a
side with ∠1. Name it ∠PVB. Is there more than one way to sketch ∠PVB? If so,
sketch all the different possibilities.
Q
14.If m∠MQV = 90, which expression can you use to find m∠VQP?
A.m∠MQP - 90
C.m∠MQP + 90
B.90 - m∠MQV D.90 + m∠VQP
P
V
M
N
Analyze Mathematical Relationships (1)(F) Find the angle measure
of the hands of a clock at each time.
15.6:00
16.7:00
17.11:00
18.4:40
19.5:20
20.2:15
21.Use Representations to Communicate
Mathematical Ideas (1)(E) According to legend,
King Arthur and his knights sat around the
Round Table to discuss matters of the kingdom.
The photo shows a round table on display at
Winchester Castle, in England. From the center
of the table, each section has the same degree
measure. If King Arthur occupied two of these
sections, what is the total degree measure of
his section?
Use a protractor. Measure and classify each angle.
22.
23.
24.
20
25.Connect Mathematical Ideas (1)(F) Your classmate constructs an angle. Then
he constructs a ray from the vertex of the angle to a point in the interior of the
angle. He measures all the angles formed. Then he moves the interior ray as
shown below. What postulate do the two pictures support?
105
105
81
63
24
42
Use the diagram at the right for Exercises 26 and 27.
Solve for x. Find the angle measures to check your work.
B
A
C
26.m∠AOB
= 4x - 2, m∠BOC = 5x + 10, m∠COD
= 2x + 14
O
D
27.m∠AOB = 28, m∠BOC = 3x - 2, m∠AOD = 6x
28.If m∠ABD = 79, what are m∠ABC and m∠DBC?
D
29.∠RQT is a straight angle. What
are m∠RQS and m∠TQS?
S
C
(6x 20)
(5x 4)
(8x 3)
B
(2x 4)
Q
T
A
R
TEXAS Test Practice
30.Two adjacent angles form a straight angle. If the larger angle is four times the
size of the smaller angle, what is the measure of the smaller angle?
A.36
B.45
C.54
D.60
31.XY has endpoints X = -72 and Y = 43. What is XY?
F.-115
G. -29
H.29
32.Use the figure at the right.
a.What is the value of x?
J.115
6x 2
A
9x 10
B
C
b.What is AC?
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1-4 Exploring Angle Pairs
TEKS FOCUS
VOCABULARY
Foundational to TEKS (6)(A) Verify
theorems about angles formed by the
intersection of lines and line segments,
including vertical angles, and angles
formed by parallel lines cut by a transversal
and prove equidistance between the
endpoints of a segment and points on its
perpendicular bisector and apply these
relationships to solve problems.
TEKS (1)(D) Communicate mathematical
ideas, reasoning, and their implications
using multiple representations, including
symbols, diagrams, graphs, and language as
appropriate.
•Adjacent angles – Adjacent angles
are two coplanar angles with a
common side, a common vertex,
and no common interior points.
•Angle bisector – An angle bisector
is a ray that divides an angle into
two congruent angles.
•Complementary angles –
Complementary angles are two
angles whose measures have a
sum of 90.
•Linear pair – A linear pair is a
pair of adjacent angles whose
noncommon sides are opposite
rays.
•Supplementary angles –
Supplementary angles are two
angles whose measures have a
sum of 180.
•Vertical angles – Vertical angles
are two angles whose sides are
opposite rays.
explicitly stated
Special angle pairs can help you identify geometric relationships. You can use these
angle pairs to find angle measures.
Key Concept Types of Angle Pairs
Definition
Example
Adjacent angles are two coplanar angles
with a common side, a common vertex,
and no common interior points.
∠1 and ∠2, ∠3 and ∠4
Vertical angles are two angles whose
∠1 and ∠2, ∠3 and ∠4
1
1
sides are opposite rays.
Complementary angles are two angles
whose measures have a sum of 90.
Each angle is called the complement
of the other.
Supplementary angles are two angles
whose measures have a sum of 180.
Each angle is called the supplement
of the other.
22
∠1 and ∠2, ∠A and ∠B
3
2
3
4
4
2
1
47
43
2
B
A
∠3 and ∠4, ∠B and ∠C
3 4
137
C
Lesson 1-4 Exploring Angle Pairs
Concept Summary Finding Information From a Diagram
There are some relationships you can assume to be true from a diagram that has
no marks or measures. There are other relationships you cannot assume directly.
For example, you can conclude the following from an unmarked diagram.
• Angles are adjacent.
• Angles are adjacent and supplementary.
• Angles are vertical angles.
You cannot conclude the following from an unmarked diagram.
• Angles or segments are congruent.
• An angle is a right angle.
• Angles are complementary.
Postulate 1-9 Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
Problem 1
Identifying Angle Pairs
What should you look
for in the diagram?
For part (A), check
whether the angle pair
matches every part of
the definition of adjacent
angles.
B
Use the diagram at the right. Is the statement true? Explain.
A jBFD and jCFD are adjacent angles.
>
o. They have a common side (FD )
N
and a common vertex (F), but they also
have common interior points.
So ∠BFD and ∠CFD are not adjacent.
B
62
common
interior
points
C
F
28
A
E
C
F
118
D
D
B jAFB and jEFD are vertical angles.
>
>
No.> FA and> FD are opposite rays, but
FE and FB are not. So ∠AFB and ∠EFD
are not vertical angles.
B
not opposite
A
rays
F
D
E
C jAFE and jBFC are complementary.
Yes. m∠AFE + m∠BFC = 62 + 28 = 90.
The sum of the angle measures is 90, so
∠AFE and ∠BFC are complementary.
B
28
A
62
E
F
C
The sum of the
measures is 90.
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Problem 2
Making Conclusions From a Diagram
How can you get
information from
a diagram?
Look for relationships
between angles. For
example, look for
congruent angles and
adjacent angles.
What can you conclude from the information in the diagram?
2
1
• ∠1 ≅ ∠2 by the markings.
• ∠3 and ∠5 are vertical angles.
3
5 4
• ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠5, and ∠5 and ∠1
are adjacent angles.
Problem 3
Finding Missing Angle Measures
Draw a diagram. Use
the definition of
supplementary angles
to write and solve an
equation.
Algebra jKPL and jJPL are a linear pair, mjKPL = 2x + 24, and
mjJPL = 4x + 36. What are the measures of jKPL and jJPL?
Step 1
m∠KPL + m∠JPL = 180
(2x + 24) + (4x + 36) = 180
6x + 60 = 180
6x = 120
x = 20 Def. of supplementary angles
L
Substitute.
Combine like terms.
(2x 24) (4x 36)
Subtract 60 from each side.
K
P
J
Step 2 Evaluate the original expressions for x = 20.
m∠KPL = 2x + 24 = 2
# 20 + 24 = 40 + 24 = 64 m∠JPL = 4x + 36 = 4 # 20 + 36 = 80 + 36 = 116
Substitute 20 for x.
Problem 4
Using an Angle Bisector to Find Angle Measures
>
Multiple Choice AC bisects jDAB. If mjDAC = 58, what is mjDAB?
29
Draw a diagram to help you
visualize what you are
given and what you need
to find.
D
58
A
C
58
87
Draw a diagram.
B
m∠CAB = m∠DAC
= 58
Definition of angle bisector
Substitute.
m∠DAB
= m∠CAB + m∠DAC
Angle Addition Postulate
= 58 + 58
Substitute.
= 116
Simplify.
The measure of ∠DAB is 116. The correct choice is D.
24
116
HO
ME
RK
O
NLINE
WO
Use the diagram at the right. Is each statement true? Explain.
1.
∠1 and ∠5 are adjacent angles.
5 1
3 2
4
2.
∠3 and ∠5 are vertical angles.
3.
∠3 and ∠4 are complementary.
Ideas (1)(D) Name an angle or angles in the diagram described by
each of the following.
A
B
4.
supplementary to ∠AOD
E
5.
adjacent and congruent to ∠AOE
60
6.
supplementary to ∠EOA
D
7.
a pair of vertical angles
O
C
Analyze Mathematical Relationships (1)(F) Find the measure of each
angle in the angle pair described.
8.
The measure of one angle is twice the measure of its supplement.
9.
The measure of one angle is 20 less than the measure of its complement.
In the diagram at the right, mjACB = 65. Find each of the following.
10.m∠ECD
F
A
11.m∠ACE
12.Analyze Mathematical Relationships (1)(F) ∠RQS and ∠TQS
are a linear pair where m∠RQS = 2x + 4 and m∠TQS = 6x + 20.
E
C
B
D
a.Solve for x.
b.Find m∠RQS and m∠TQS.
c.Show how you can check your answer.
13.Justify Mathematical Arguments (1)(G) In the diagram at the right, are
∠1 and ∠2 adjacent? Justify your reasoning.
1
>
2
14.Explain Mathematical Ideas (1)(G) When BX bisects ∠ABC, ∠ABX ≅ ∠CBX .
One student claims there is always a related equation m∠ABX = 12 m∠ABC.
Another student claims the related equation is 2m∠ABX = m∠ABC.
Who is correct? Explain.
hsm11gmse_0105_t0
15.∠EFG and ∠GFH are a linear pair,
m∠EFG = 2n + 21, and m∠GFH = 4n + 15.
What are m∠EFG and m∠GFH?
>
F
16.In the diagram, GH bisects ∠FGI .
a.Solve for x and find m∠FGH.
b.Find m∠HGI .
c.Find m∠FGI .
(3x 3)
G
H
(4x 14)
I
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STEM
17.Apply Mathematics (1)(A) A beam of light and a
mirror can be used to study the behavior of light.
Light that strikes the mirror is reflected so that the
angle of reflection and the angle of incidence are
congruent. In the diagram, ∠ABC has a measure of 41.
a.Name the angle of reflection and find its measure.
b.Find m∠ABD.
c.Find m∠ABE and m∠DBF .
18.Connect Mathematical Ideas (1)(F) Describe all
situations where vertical angles are also supplementary.
>
>
Angle of
reflection
Angle of
incidence
C
A
D
E
F
B
>
19.XC bisects> ∠AXB, XD bisects> ∠AXC, XE bisects
∠AXD,
XF bisects ∠EXD, XG bisects ∠EXF , and
>
XH bisects ∠DXB. If m∠DXC = 16, find m∠GXH.
Ideas (1)(D) For Exercises 20–23, can you make each conclusion
from the information in the diagram? Explain.
E
F
A
20.∠J ≅ ∠D
21.C is the midpoint of JD.
22.∠JAE and ∠EAF are adjacent and supplementary.
J
23.∠EAF and ∠JAD are vertical angles.
C
D
24.Connect Mathematical Ideas (1)(F) The x- and y-axes of the coordinate plane
form four right angles. The interior of each of the right angles is a quadrant of
the coordinate plane. What is the equation for the line that containshsm11gmse_0105_t00907.ai
the
angle bisector of these axes and lies in Quadrants I and III?
TEXAS Test Practice
25.Which statement is true?
A.A right angle has a complement.
B.An obtuse angle has a complement.
C.The supplement of a right angle is a right angle.
D.Every angle has a supplement.
26.The diagram shows distance in meters. How far, in
meters, is it from the parking lot to your house?
F.
44
H.183
G.135
J.189
P
27
0
26
162
1-5 Basic Constructions
TEKS FOCUS
VOCABULARY
TEKS (5)(B) Construct congruent
segments, congruent angles, a segment
bisector, an angle bisector, perpendicular
lines, the perpendicular bisector or a line
segment, and a line parallel to a given
line through a point not on a line using a
compass and a straightedge.
•Compass – A compass is a
TEKS (1)(E) Create and use
representations to organize, record, and
communicate mathematical ideas.
•Perpendicular bisector – A
geometric tool used to draw
circles and parts of circles
called arcs.
•Construction – A construction is
a geometric figure drawn using a
straightedge and a compass.
•Perpendicular lines – Perpendicular
lines are two lines that intersect to
form right angles.
•Straightedge – A straightedge is a
ruler with no markings on it.
perpendicular bisector of a
segment is a line, segment, or
ray that is perpendicular to the
segment at its midpoint.
You can use special geometric tools to make a figure that is congruent to an original
figure without measuring. This method is more accurate than sketching and drawing.
Problem 1
TEKS Process Standard (1)(E)
Constructing Congruent Segments
Use a straightedge and compass to construct a segment congruent
to a given segment.
Given: AB
A
Construct: CD so that CD ≅ AB
Why must the
compass setting stay
the same?
Using the same compass
setting keeps segments
congruent. It guarantees
that the lengths of AB
and CD are exactly the
same.
Step 1
Use a straightedge to draw a ray with endpoint C.
Step 2
Open the compass to the length of AB.
C
A
Step 3With the same compass setting, put the compass point
on point C. Draw an arc that intersects the ray. Label
the point of intersection D.
CD ≅ AB
B
B
C
D
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Problem 2
Constructing Congruent Angles
Use a straightedge and compass to construct an angle congruent to a given angle.
Given: ∠A
A
Construct: ∠S so that ∠S ≅ ∠A
Step 1
Use a straightedge to draw a ray with endpoint S.
S
Step 2
Why do you need
points like B and C?
B and C are reference
points on the original
angle. You can construct
a congruent angle by
locating corresponding
points R and T on your
new angle.
With the compass point on vertex A, draw an arc that intersects the
sides of ∠A. Label the points of intersection B and C.
B
Ahsm11gmse_0106_t00923.ai
C
Step 3
With the same compass setting, put the compass point on point S.
Draw an arc and label its point of intersection with the ray as R.
S
R
Step 4
T
Open the compass to the length BC. Keeping the same compass
setting, put the compass point on R. Draw an arc to locate point T.
Shsm11gmse_0106_t00925.ai
R
Step 5
T
>
Use a straightedge to draw ST .
S
R
∠S ≅ ∠A
Problem 3
What is a bisector of
a line segment?
A bisector of a line
segment is a line,
segment, or ray that
passes through the
midpoint of the segment.
(It can also be the
midpoint itself.)
Constructing a Segment Bisector
Use a compass and straightedge to construct a bisector of a
given segment.
Given: AB
<
>
<
>
Construct: CD so that CD is a bisector of AB
A
B
Step 1
With the compass point on endpoint A, draw an arc above
the segment. Then use the same compass setting to draw a
similar arc, with point B as the center, below the line segment.
A
B
28
Lesson 1-5 Basic Constructions
Problem 3
continued
Step 2
Open the compass to a setting greater that the one used in
Step 1. With the compass point on endpoint A, draw an arc
which intersects the arc below the segment. Then use the
same compass setting to draw a similar arc, with point B as
the center, which intersects the arc above the segment.
A
B
Step 3
Label the intersection of the arcs above AB as C. Label the
intersection of the arcs below AB as D. Use a straightedge
to draw a line through C and D.
C
< >
CD bisects AB.
A
B
D
Problem 4
TEKS Process Standard (1)(E)
Constructing the Perpendicular Bisector
Use a straightedge and compass to construct the perpendicular
bisector of a segment.
Given: AB
< >
A
B
< >
Construct: XY so that XY is the perpendicular bisector of AB
Why must the
compass opening be
greater than 12 AB?
If the opening is less than
1
2 AB, the two arcs will
not intersect in Step 2.
Step 1Put the compass point on point A and draw a long
arc as shown. Be sure the opening is greater than 12AB.
Step 2With the same compass setting, put the compass point on
point B and draw another long arc. Label the points where
the two arcs intersect as X and Y.
< >
Step 3
Use a straightedge
to draw XY . Label the point of intersection
< >
of AB and XY as M, the midpoint of AB.
< >
< >
XY # AB at midpoint M, so XY is the perpendicular
bisector of AB.
hsm11gmse_0106_t00934.a
A
B
X
Ahsm11gmse_0106_t00935
B
Y
X
hsm11gmse_0106_t00936
A
B
M
Y
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hsm11gmse_0106_t00937
Problem 5
Constructing the Angle Bisector
Use a straightedge and compass to construct the bisector of an angle.
A
Given: ∠A
>
Construct: AD , the bisector of ∠A
B
Step 1
Put the compass point on vertex A. Draw an arc that intersects
the sides of ∠A. Label the points of intersection B and C.
A
C
B
Step 2
NLINE
HO
ME
RK
O
Why must the arcs
intersect?
The arcs need to intersect
so that you have a point
through which to draw
a ray.
WO
Put the compass point on point C and draw an arc in the interior of ∠A.
With the same compass setting, draw an arc using point B. Be sure the
arcs intersect. Label the point where the two arcs intersect as D.
A
C
Step 3
B
>
Use a straightedge to draw AD .
>
AD is the bisector of ∠CAB.
A
C
Sketch the figure described. Explain how to construct it. Then do the construction.
< >
< >
>
2.ST bisects right ∠PSQ.
3.
Connect Mathematical Ideas (1)(F) How is constructing an angle bisector similar
to constructing a perpendicular bisector?
4.
Explain Mathematical Ideas (1)(G) Explain how to do each construction with a
compass and straightedge.
a.Draw a segment PQ. Construct the midpoint of PQ.
b.Divide PQ into four congruent segments.
5.a. Draw a large triangle with three acute angles. Construct the bisectors of the
three angles. What appears to be true about the three angle bisectors?
b.Repeat the constructions with a triangle that has one obtuse angle.
c.What appears to be true about the three angle bisectors of any triangle?
Use a ruler to draw segments of 2 cm, 4 cm, and 5 cm. Then construct each triangle
with the given side measures, if possible. If it is not possible, explain why not.
6.
4 cm, 4 cm, and 5 cm
30
D
1.
XY # YZ D
7.2 cm, 2 cm, and 4 cm
8.
Two triangles are congruent if each side and each angle of one triangle is
congruent to a side or angle of the other triangle. In Topic 4, you will learn
that if each side of one triangle is congruent to a side of the other triangle,
then you can conclude that the triangles are congruent without finding
the angles. Explain how you can use congruent triangles to justify the
angle bisector construction.
9.
Answer the questions about a segment in a plane. Explain each answer.
a.How many midpoints does the segment have?
hsm11gmse_0106_t00953
b.How many bisectors does it have?
c.How many lines in the plane are its perpendicular bisectors?
d.How many lines in space are its perpendicular bisectors?
10.Study the figures. Complete the definition of a line
perpendicular to a plane: A line is perpendicular to a
plane if it is ? to every line in the plane that ? .
t
r
M
P
11.a. Use your compass to draw a circle. Locate three
points A, B, and C on the circle.
b.Draw AB and BC. Then construct the
perpendicular bisectors of AB and BC.
Line r plane M.
Line t is not plane P.
c.Analyze Mathematical Relationships (1)(F) Label the intersection of the two
perpendicular bisectors as point O. What do you think is true about point O?
hsm11gmse_0106_t00952
Create Representations to Communicate Mathematical Ideas (1)(E) For
Exercises 12–16, draw a diagram similar to the given one. Then do the construction.
Check your work with a ruler or a protractor.
12.Construct XY congruent to AB.
13.Construct VW so that VW = 2AB.
14.Construct ∠D so that ∠D ≅ ∠C.
15.Construct ∠F so that m∠F = 2m∠C.
16.Construct the perpendicular bisector of AB.
A
B
C
17.Draw acute ∠PQR. Then construct its bisector.
18.Draw obtuse ∠XQZ. Then construct its bisector.
19.Display Mathematical Ideas (1)(G) Draw an ∠A. Construct an angle
whose
measure is 14m∠A.
20.a. Draw a segment, XY . Construct a triangle with sides congruent to XY .
b.Measure the angles of the triangle.
c.Describe how to construct a 60° angle using what you know. Then describe
how to construct a 30° angle.
PearsonTEXAS.com
31
21.Which steps best describe how to construct the pattern at the right?
A.Use a straightedge to draw the segment and then a compass to draw five half circles.
B.Use a straightedge to draw the segment and then a compass to draw six half circles.
C.Use a compasshsm11gmse_0106_t00951.ai
to draw five half circles and then a straightedge to join their ends.
D.Use a compass to draw six half circles and then a straightedge to join their ends.
For Exercises 22–24, determine if the steps construct a segment bisector for a given
line segment AB.
22.Use a straightedge and compass to construct two perpendicular lines to AB, one
through point A and another through point B. Open the compass to a certain length
and draw an arc, with point A as the center, intersecting one perpendicular above
the segment. With the same compass setting, draw an arc, with point B as the
center, intersecting the other perpendicular below the segment. Use a straightedge
to draw a straight line through both intersection points.
23.Use a compass to construct an arc centered at point A above the segment, with
radius greater than one-half AB. Using a straightedge, draw a line through point A
intersecting the arc above the segment. Open the compass to the distance from this
intersection point to point B. Draw an arc with center B that intersects AB. Use a
straightedge to draw a line through both intersection points.
24.Use a compass to construct an arc centered at point A above the segment, with length
greater than one-half AB. With the same compass setting, draw an arc centered at
point B, intersecting the first arc. Repeat this procedure, but below the segment. Use a
straightedge to draw a straight line through both intersection points.
TEXAS Test Practice
25.Given the diagram at the right, what is NOT a reasonable name for the angle?
A.∠ABC
C.∠CBA
B.∠B
D.∠ACB
A
B
C
26.What must you do to construct the midpoint of a segment?
F.
Measure half its length.
G.Construct an angle bisector. H.Measure twice its length.
J.Construct a perpendicular bisector.
2
27.M is the midpoint of XY . Find the value of x. Show your work.
x 2
X
x
M
Y
32
Technology Lab
Use With Lesson 1-5
Exploring Constructions
teks (5)(A), (1)(G)
You can use Draw tools or Construct tools in geometry software to make points, lines,
and planes. A figure made by Draw has no constraints. When you manipulate, or try
to change, a figure made by Draw, it moves or changes size freely. A figure made by
Construct is related to an existing object. When you manipulate the existing object,
the constructed object moves or resizes accordingly.
In this Activity, you will explore the difference between Draw and Construct.
<
>
Draw AB and Construct the perpendicular
< > bisector DC . Then Draw EF and
Construct G, any point on EF . Draw HG .
D
C
A
H
B
F
E
G
1.Find EG, GF, and m∠HGF . Try to drag G so that EG = GF . Try to drag H
so that m∠HGF = 90. Were you able to draw the perpendicular bisector
of EF ? Explain.
< >
hsm11gmse_0106b_t00983.ai
2.Drag A and B. Observe AC, CB, and m∠DCB. Is DC always the
perpendicular bisector of AB no matter how you manipulate the figure?
3.Drag E and F. Observe
< > EG, GF, and m∠HGF . How is the relationship < >
between EF and HG different from the relationship between AB and DC ?
4.Write a description of the general difference between Draw and Construct.
Then
between EF and
< > use your description to explain why the relationship
< >
HG differs from the relationship between AB and DC .
PearsonTEXAS.com
33
Technology Lab
Exercises
>
5.a.Draw ∠NOP. Draw OQ in the interior of ∠NOP. Drag Q until
m∠NOQ = m∠QOP.
N Q
P
O
b.
Manipulate
the figure and observe the different angle measures.
>
Is OQ always the angle bisector of ∠NOP?
>
6.a.Draw ∠JKL and Construct its angle bisector, KM .
J
M
K
L
b.
Manipulate
the figure and observe the different angle measures.
>
Is KM always the angle bisector of ∠JKL?
c.
How can you manipulate the figure on the screen so that it shows
a right angle? Justify your
answer.
34
Technology Lab Exploring Constructions
continued
Activity Lab
Use With Lesson 1-5
Classifying Polygons
teks (1)(F)
In geometry, a figure that lies in a plane is called a plane figure.
A polygon is a closed plane figure formed by three or more segments. Each segment
intersects exactly two other segments at their endpoints. No two segments with a
common endpoint are collinear. Each segment is called a side. Each endpoint of a
side is a vertex.
B
A
B
B
A
A
C
E
C
C
E
D
A polygon
D
D
Not a polygon;
not a closed figure
Not a polygon;
two sides intersect
between endpoints.
hsm11gmse_0108a_t01052.ai
hsm11gmse_0108a_t01050.ai hsm11gmse_0108a_t01051.ai
To name a polygon, start at any vertex and list the vertices consecutively
in a clockwise or counterclockwise direction.
Example 1
D
Name the polygon. Then identify its sides and angles.
Two names for this polygon are DHKMGB and MKHDBG.
E
H
B
K
sides: DH, HK , KM, MG, GB, BD
angles: ∠D, ∠H, ∠K, ∠M, ∠G, ∠B
G
M
You can classify a polygon by its number of sides. The tables below show the names
of some common polygons.
Names of Common Polygons
Name
Sides
Name
Triangle, or trigon
9
Nonagon, or enneagon
4
Quadrilateral, or tetragon
10
Decagon
5
Pentagon
11
Hendecagon
6
Hexagon
12
Dodecagon
7
Heptagon
8
Octagon
n
…
3
…
Sides
n-gon
PearsonTEXAS.com
35
Activity Lab
continued
You can also classify a polygon as concave or convex, using the diagonals of the
polygon. A diagonal is a segment that connects two nonconsecutive vertices.
A
D
Y
R
P
M
K
W
Q
G
S
T
A convex polygon has no diagonal
with points outside the polygon.
A concave polygon has at least one
diagonal with points outside the polygon.
In this textbook, a polygon
is convex unless otherwise stated.
Example 2
Classify the polygon by its number of sides. Tell whether the polygon is convex
or concave.
The polygon has six sides. Therefore, it is a hexagon.
No diagonal of the hexagon contains points outside the hexagon. The hexagon is convex.
Exercises
Is the figure a polygon? If not, explain why.
1.
2.
3.
4.
Name the polygon. Then identify its sides and angles.
5.
M
K
6.
7.
C
P T
E
X
W
F
A
P
B
H
L
G
N
Classify the polygon by its number of sides. Tell whether the polygon is convex
or concave.
8.
36
Activity
Lab Classifying Polygons
9.
10.
Topic 1 Review
TOPIC VOCABULARY
• acute, right, obtuse, straight
angles, p. 17
• congruent angles, p. 16
• measure of an angle, p. 17
• sides of an angle, p. 16
• congruent segments, p. 10
• midpoint, p. 11
• space, p. 4
• adjacent angles, p. 22
• construction, p. 27
• perpendicular bisector, p. 27
• straightedge, p. 27
• angle, p. 16
• coordinate, p. 10
• perpendicular lines, p. 27
• supplementary • angle bisector, p. 22
• coplanar, p. 4
• point, line, plane, p. 4
• collinear points, p. 4
• distance, p. 10
• postulate, axiom, p. 4
• vertex of an angle, p. 16
• complementary angles, p. 22
• diagonal, p. 36
• ray, opposite rays, p. 5
• vertical angles, p. 22
• compass, p. 27
• intersection, p. 4
• segment, p. 5
• concave, convex, p. 36
• linear pair, p. 22
• segment bisector, p. 11
angles, p. 22
Check Your Understanding
Choose the correct term to complete each sentence.
1.A ray that divides an angle into two congruent angles is a(n) ? .
2. ? are two lines that intersect to form right angles.
3.A(n) ? is a geometric figure drawn using a straightedge and a compass.
1-1 Points, Lines, and Planes
Quick Review
Exercises
A point indicates a location and has no size.
A line is represented by a straight path that extends in two
opposite directions without end and has no thickness.
A plane is represented by a flat surface that extends
without end and has no thickness.
Points that lie on the same line are collinear points.
Points and lines in the same plane are coplanar.
Segments and rays are parts of lines.
Use the figure below for Exercises 4–6.
Example
Segments: AB, AC, BC, and BD
5.Name the intersection of
planes QRB and TSR.
6.Name three noncollinear
points.
T
S
Q
R
D
A
C
B
Determine whether the statement is true or false.
Explain your reasoning.
7.Two points are always collinear.
hsm11gmse_01cr_t01238.ai
D
Name all the segments and rays in
the figure.
4.Name two intersecting lines.
>
>
8.LM and ML are the same ray.
>
>
>
>
>
>
>
Rays: BA , CA or CB , AC or AB , BC , and BD
A
B
C
PearsonTEXAS.com
37
1-2 Measuring Segments
Quick Review
Exercises
The distance between two points is the length of the
segment connecting those points. Segments with the same
length are congruent segments. A midpoint of a segment
divides the segment into two congruent segments.
For Exercises 9 and 10, use the number line below.
P
4
H
2
0
2
4
Example
9.Find two possible coordinates of Q such that
PQ = 5.
Are AB and CD congruent?
10. Find the coordinate
of the midpoint of PH.
C
8
A D
6
4
B
0
2
2
11. Find the value of m.
3m 5
AB = 0 -3 - 2 0 = 0 -5 0 = 5
4m 10
A
CD = 0 -7 - ( -2) 0 = 0 -5 0 = 5
AB = CD, sohsm11gmse_01cr_t01239.ai
AB ≅ CD.
B
C
12. If XZ = 50, what are XY and YZ?
a
X
1-3 Measuring Angles
a8
Z
Y
Quick Review
Exercises
Two rays with the same endpoint form an angle. The
endpoint is the vertex of the angle. You can classify angles
as acute, right, obtuse, or straight. Angles with the same
measure are congruent angles.
Classify each angle as acute, right, obtuse, or straight.
13. 14.
Example
If mjAOB = 47 and mjBOC = 73, find mjAOC.
B
A
Use the diagram below for Exercises 15 and 16.
N
M
P
O
C
m∠AOC = m∠AOB + m∠BOC
= 47 + 73
= 120
38
Topic 1 Review
Q
R
15. If m∠MQR = 61 and m∠MQP = 25, find m∠PQR.
16. If m∠NQM = 2x + 8 and m∠PQR = x + 22,
find the value of x.hsm11gmse_01cr_t01247.ai
1-4 Exploring Angle Pairs
Quick Review
Exercises
Some pairs of angles have special names.
Name a pair of each of the following.
• A
djacent angles: coplanar angles with a common side,
a common vertex, and no common interior points
17. complementary angles
• Vertical angles: sides are opposite rays
18. supplementary angles
• Complementary angles: measures have a sum of 90
19. vertical angles
• Supplementary angles: measures have a sum of 180
20. linear pair
• L inear pair: adjacent angles with noncommon sides as
opposite rays
Find the value of x.
Angles of a linear pair are supplementary.
21.
(3x 31)
B
A
D
E
C
F
(2x 6)
Example
E
Are jACE and jBCD vertical
angles? Explain.
A
No. They have only one set of sides
with opposite rays.
22.
B
C
3x
(4x 15)
D
1-5 Basic Constructionshsm11gmse_01cr_t01248.ai
Quick Review
Exercises
Construction is the process of making geometric
figures using a compass and a straightedge. Four basic
constructions involve congruent segments, congruent
angles, and bisectors of segments and angles.
23. Use a protractor to draw a 73° angle. Then construct an
angle congruent to it.
Example
Construct AB congruent to EF .
E
F
24. Use a protractor to draw a 60° angle. Then construct the
bisector of the angle.
25. Sketch LM on paper. Construct a line segment
congruent to LM. Then construct the perpendicular
bisector of your line segment.
Step 1
Draw a ray with endpoint A.
Step 2
Open the compass to the length of
EF . Keep that compass setting and
put the compass point on point A.
Draw an arc that intersects the ray.
Label the point of intersection B.
M
L
A
26. a.Sketch ∠B on paper. Construct
an angle congruent to ∠B.
b. Construct the bisector of your
angle from part (a).
B
B
A
hsm11gmse_01cr_t01256.
PearsonTEXAS.com
39
Topic 1 TEKS Cumulative Practice
Multiple Choice
Read each question. Then write the letter of the correct
answer on your paper.
1.Points A, B, C, D, and E are collinear. A is to the right of
B, E is to the right of D, and B is to the left of C. Which
of the following is NOT a possible arrangement of the
points from left to right?
A.
D, B, A, E, CC.
B, D, E, C, A
5.Which construction requires drawing only one arc
with a compass?
A.
constructing congruent segments
B.
constructing congruent angles
C.
constructing the perpendicular bisector
D.
constructing the angle bisector
< >
6.What is the intersection of AC and plane Q?
B.
D, B, A, C, ED.
B, A, E, C, D
2.Which postulate most closely resembles the Angle
Addition Postulate?
F.
Ruler Postulate
A
E
B
D
Q
C
G.
Protractor Postulate
H.
Segment Addition Postulate
F.
point E H.
point B
J.
Area Addition Postulate
G.
point Q J.
point D
3.What is the coordinate of the point that is 14 the distance
from 20 to 4 on a number line?
7.If ∠A and ∠B are supplementary angles, what angle
relationship between ∠A and ∠B CANNOT be true?
A.
8C.
16
A.
∠A and ∠B are right angles.
B.
9D.
15
B.
∠A and ∠B are adjacent angles.
4.Given: ∠A
C.
∠A and ∠B are complementary angles.
What is the second step in constructing the
angle bisector of ∠A?
D.
∠A and ∠B are congruent angles.
B
D
A
F.
plane
G.
angle bisector
H.
construction
C
J.
opposite rays
>
F.
Draw AD .
G.
From points B and C, use the same compass setting
hsm11gmse_01cu_t01227.ai
to draw arcs that intersect at D.
H.
Draw a line segment connecting points B and C.
J.
From point A, draw an arc that intersects the sides of
the angle at points B and C.
40
8.Which geometric term is undefined?
Topic 1 TEKS Cumulative Practice
9.The measure of an angle is 12 less than twice the
measure of its supplement. What is the measure of
the angle?
A.
28C.
64
B.
34D.
116
10. If m∠BDJ = 7y + 2 and m∠JDR = 2y + 7, find the value
of y.
B
K
J
D
R
Constructed Response
18. Copy the graph below. Find the midpoints of
two adjacent sides of the square. Connect the
perpendicular bisectors of the two adjacent sides.
What is the perimeter of the new square? Show
your work.
y
F.
y=3
G.
y=8
4
H.
y=5
2
J.
y=9
x
6 4
11. Which statement is true?
2
O
A.
It is possible for three points to be noncoplanar.
2
B.
A plane containing two points of a line contains the
entire line.
4
C.
Complementary angles are congruent.
D.
A straight angle has a supplement.
12. The measure of an angle is 78 less than the measure of
its complement. What is the measure of the angle?
13. The measure of an angle is one third the measure of its
supplement. What is the measure of the angle?
14. Y is the midpoint of XZ. What is the value of b?
2b 1
26 4b
Y
19. Suppose PQ = QR. Your friend says that Q is always
the midpoint of PR. Is he correct? Explain.
20. Why
might it be useful to have more than one way to
name an angle?
Gridded Response
X
4
Z
15. The sum of the measures of a complement and a
supplement of an angle is 200. What is the measure of
the angle?
21. The bisector of obtuse ∠AOD goes through point C.
The bisector of ∠AOC goes through point B.
a.If m∠COD = 4x + 12, what is the measure of
∠BOC in terms of x? How do you know?
b.If m∠COD = 4x + 12 and m∠AOD = 120, what is
the value of x? How do you know?
22. In JK , JH = 4x - 15 and HK = 2x + 3, where H is
between J and K on JK .
a.If JK = 48, find the value of x.
b.Is H the midpoint of JK ? Explain.
16. In AB, the coordinate of point A is -4, and the
coordinate of point B is 6. What is the coordinate of a
point C such that C is 38 of the distance from point A to
point B?
<
>
17. VW is the bisector of AY , and they intersect at E. If
EY = 3.5, what is AY ?
PearsonTEXAS.com
41

Homework Helper Chapter 1

Transcription

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