Download 3 Maintaining Mathematical Proficiency Chapter

Name _________________________________________________________ Date _________
Chapter
3
Maintaining Mathematical Proficiency
Write an equation in point-slope form of the line that passes through the given
point and has the given slope.
1.
(2, 1); m
4.
(− 6, 0); m
= 3
= −1
2
= 2
2.
(3, 5); m
5.
(2, − 7); m
3
= 6
3.
(0, 4); m
6.
(− 4, − 7); m
= −5
= −1
Write an equation in slope-intercept form of the line that passes through the given
point and has the given slope.
7.
10.
64
= 3
8.
(− 4, 3); m
= 1
9.
= −10
11.
(− 5, 0); m
= 7
12.
(0, − 8); m
(1, 9); m
5
Geometry
Student Journal
3
(2, −1); m
= 5
(−3, − 2); m
= −2
5
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.1
Date __________
Pairs of Lines and Angles
For use with Exploration 3.1
Essential Question What does it mean when two lines are parallel,
intersecting, coincident, or skew?
1
EXPLORATION: Points of Intersection
Work with a partner. Write the number of points of intersection of each pair of
coplanar lines.
a. parallel lines
_________________
2
b. intersecting lines
c. coincident lines
_________________
_________________
EXPLORATION: Classifying Pairs of Lines
Work with a partner. The figure shows a right rectangular
prism. All its angles are right angles. Classify each of the
following pairs of lines as parallel, intersecting, coincident,
or skew. Justify your answers. (Two lines are skew lines
when they do not intersect and are not coplanar.)
B
a.


AB and BC
b.


AD and BC
c.


EI and IH
d.


BF and EH
e.


EF and CG
f.


AB and GH
Copyright © Big Ideas Learning, LLC
All rights reserved.
Classification
D
A
F
E
Pair of Lines
C
G
I
H
Reason
Geometry
Student Journal
65
Name _________________________________________________________ Date _________
3.1
3
Pairs of Lines and Angles (continued)
EXPLORATION: Identifying Pairs of Angles
Work with a partner. In the figure, two parallel lines are
intersected by a third line called a transversal.
a. Identify all the pairs of vertical angles. Explain your
reasoning.
5 6
1 2
8 7
4 3
b. Identify all the linear pairs of angles. Explain your reasoning.
Communicate Your Answer
4. What does it mean when two lines are parallel, intersecting, coincident, or skew?
5. In Exploration 2, find three more pairs of lines that are different from those given.
Classify the pairs of lines as parallel, intersecting, coincident, or skew. Justify
your answers.
66
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.1
Date __________
Notetaking with Vocabulary
For use after Lesson 3.1
In your own words, write the meaning of each vocabulary term.
parallel lines
skew lines
parallel planes
transversal
corresponding angles
alternate interior angles
alternate exterior angles
consecutive interior angles
Notes:
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
67
Name _________________________________________________________ Date _________
3.1
Notetaking with Vocabulary (continued)
Core Concepts
Parallel Lines, Skew Lines, and Parallel Planes
Two lines that do not intersect are either parallel lines or skew lines. Two lines are
parallel lines when they do not intersect and are coplanar. Two lines are skew lines
when they do not intersect and are not coplanar. Also, two planes that do not intersect
are parallel planes.
Lines m and n are parallel lines ( m  n).
k
m
T
n
U
Lines m and k are skew lines.
Planes T and U are parallel planes (T  U ).
Lines k and n are intersecting lines, and there is a
plane (not shown) containing them.
Small directed arrows, as shown on lines m and n above, are used to show that lines are
parallel. The symbol  means “is parallel to,” as in m  n.
Segments and rays are parallel when they lie in parallel lines. A line is parallel to a plane
when the line is in a plane parallel to the given plane. In the diagram above, line n is
parallel to plane U.
Notes:
Postulate 3.1 Parallel Postulate
If there is a line and a point not on the line, then there is exactly
one line through the point parallel to the given line.
P
There is exactly one line through P parallel to .
Notes:
68
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.1
Date __________
Notetaking with Vocabulary (continued)
Postulate 3.2 Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly
one line through the point perpendicular to the given line.
P
There is exactly one line through P perpendicular to .
Notes:
Angles Formed by Transversals
t
2
6
t
4
5
Two angles are corresponding angles when they
have corresponding positions. For example, ∠ 2
and ∠6 are above the lines and to the right of the
Two angles are alternate interior angles when
they lie between the two lines and on opposite sides
of the transversal t.
transversal t.
t
1
8
Two angles are alternate exterior angles when
they lie outside the two lines and on opposite sides
of the transversal t.
t
3
5
Two angles are consecutive interior angles when
they lie between the two lines and on the same side
of the transversal t.
Notes:
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
69
Name _________________________________________________________ Date _________
Notetaking with Vocabulary (continued)
3.1
3.1
Extra Practice
In Exercises 1–4, think of each segment in the diagram as part of a line.
Which line(s) or plane(s) contain point B and appear to fit the description?

1. line(s) skew to FG.
F

2. line(s) perpendicular to FG.
4. plane(s) parallel to plane FGH.
E
A

3. line(s) parallel to FG.
G
H
B
D
C
In Exercises 5–8, use the diagram.
5. Name a pair of parallel lines.
U
6. Name a pair of perpendicular lines.


7. Is WX  QR ? Explain.
X
W
N
Y


Q
S
Z
T
V
R
8. Is ST ⊥ NV ? Explain.
In Exercises 9–12, identify all pairs of angles of the given type.
9. corresponding
6
8
5
7
10. alternate interior
11. alternate exterior
2 4
1
3
10
12
9
11
12. consecutive interior
70
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.2
Date __________
Parallel Lines and Transversals
For use with Exploration 3.2
Essential Question When two parallel lines are cut by a transversal,
which of the resulting pairs of angles are congruent?
1
EXPLORATION: Exploring Parallel Lines
Go to BigIdeasMath.com for an interactive tool to
investigate this exploration.
6
D
5
Work with a partner.
Use dynamic geometry software to draw two parallel
lines. Draw a third line that intersects both parallel
lines. Find the measures of the eight angles that are
formed. What can you conclude?
4
B
3
E
1
2
4 3
1
A
0
−3
2
F
6
8 7
5
2
−2
−1
0
1
2
3
4
C
5
6
EXPLORATION: Writing Conjectures
Work with a partner. Use the results of Exploration 1 to write conjectures about the
following pairs of angles formed by two parallel lines and a transversal.
a. corresponding angles
1 2
4 3
5 6
8 7
Copyright © Big Ideas Learning, LLC
All rights reserved.
b. alternate interior angles
1 2
4 3
5 6
8 7
Geometry
Student Journal
71
Name _________________________________________________________ Date _________
3.2
2
Parallel Lines and Transversals (continued)
EXPLORATION: Writing Conjectures (continued)
c. alternate exterior angles
1 2
4 3
5 6
8 7
d. consecutive interior angles
1 2
4 3
5 6
8 7
Communicate Your Answer
3. When two parallel lines are cut by a transversal, which of the resulting pairs
of angles are congruent?
4. In Exploration 2, m∠1 = 80°. Find the other angle measures.
72
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
Date __________
Notetaking with Vocabulary
3.2
For use after Lesson 3.2
In your own words, write the meaning of each vocabulary term.
corresponding angles
parallel lines
supplementary angles
vertical angles
Theorems
Theorem 3.1 Corresponding Angles Theorem
t
If two parallel lines are cut by a transversal, then the pairs of corresponding angles
are congruent.
Examples
1 2
3 4
In the diagram, ∠2 ≅ ∠6 and ∠3 ≅ ∠7.
5 6
7 8
Theorem 3.2 Alternate Interior Angles Theorem
p
q
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles
are congruent.
Examples
In the diagram, ∠3 ≅ ∠6 and ∠4 ≅ ∠5.
Theorem 3.3 Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles
are congruent.
Examples
In the diagram, ∠1 ≅ ∠8 and ∠2 ≅ ∠7.
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
73
Name _________________________________________________________ Date _________
3.2
Notetaking with Vocabulary (continued)
Theorem 3.4 Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of consecutive interior
angles are supplementary.
In the diagram, ∠3 and ∠5 are supplementary, and ∠ 4 and ∠6
are supplementary.
Examples
Notes:
t
1 2
3 4
5 6
7 8
p
q
Extra Practice
In Exercises 1–4, find m∠1 and m∠2. Tell which theorem you use in each case.
1.
110°
1
2
2.
63°
2
1
3.
1
95°
2
74
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.2
Date __________
Notetaking with Vocabulary (continued)
4.
1
2
101°
In Exercises 5–8, find the value of x. Show your steps.
5.
110°
(x + 12)°
6.
(10x − 55)°
45°
7.
4x° 6
8.
(2x − 3)°
52°
2
Copyright © Big Ideas Learning, LLC
All rights reserved.
153°
Geometry
Student Journal
75
Name _________________________________________________________ Date _________
3.3
Proofs with Parallel Lines
For use with Exploration 3.3
Essential Question For which of the theorems involving parallel lines
and transversals is the converse true?
1
EXPLORATION: Exploring Converses
Work with a partner. Write the converse of each conditional statement. Draw a
diagram to represent the converse. Determine whether the converse is true. Justify
your conclusion.
a. Corresponding Angles Theorem (Theorem 3.1)
If two parallel lines are cut by a transversal, then the pairs of
corresponding angles are congruent.
Converse
1 2
4 3
5 6
8 7
1 2
4 3
5 6
8 7
b. Alternate Interior Angles Theorem (Theorem 3.2)
If two parallel lines are cut by a transversal, then the pairs of
alternate interior angles are congruent.
Converse
c. Alternate Exterior Angles Theorem (Theorem 3.3)
If two parallel lines are cut by a transversal, then the pairs of
alternate exterior angles are congruent.
Converse
76
Geometry
Student Journal
1 2
4 3
5 6
8 7
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.3
1
Date __________
Proofs with Parallel Lines (continued)
EXPLORATION: Exploring Converses (continued)
d. Consecutive Interior Angles Theorem (Theorem 3.4)
If two parallel lines are cut by a transversal, then the pairs of
consecutive interior angles are supplementary.
1 2
4 3
Converse
5 6
8 7
Communicate Your Answer
2. For which of the theorems involving parallel lines and transversals is the
converse true?
3. In Exploration 1, explain how you would prove any of the theorems that
you found to be true.
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
77
Name _________________________________________________________ Date _________
3.3
Notetaking with Vocabulary
For use after Lesson 3.3
In your own words, write the meaning of each vocabulary term.
converse
parallel lines
transversal
corresponding angles
congruent
alternate interior angles
alternate exterior angles
consecutive interior angles
Theorems
Theorem 3.5 Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles
are congruent, then the lines are parallel.
2
6
j
k
Notes:
j  k
78
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
Name_________________________________________________________
3.3
Date __________
Notetaking with Vocabulary (continued)
Theorem 3.6 Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
j
4
5
k
Notes:
j  k
Theorem 3.7 Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior angles are
congruent, then the lines are parallel.
1
j
k
8
Notes:
j  k
Theorem 3.8 Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive interior angles are
supplementary, then the lines are parallel.
3
j
5
Notes:
k
If ∠3 and ∠5 are
supplementary, then j  k .
Theorem 3.9 Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
p
q
r
Notes:
If p  q and q  r , then
p  r.
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
79
Name _________________________________________________________ Date _________
3.3
Notetaking with Vocabulary (continued)
Extra Practice
In Exercises 1 and 2, find the value of x that makes m  n. Explain your reasoning.
1.
2.
m
n
95°
m
(8x + 55)°
130°
(200 − 2x)°
n
In Exercises 3–6, decide whether there is enough information to prove that m  n.
If so, state the theorem you would use.
3.
m
n
4.
r
r
m
n
5.
r
6.
s
m
m
n
80
Geometry
Student Journal
s
n
r
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.4
Date __________
Proofs with Perpendicular Lines
For use with Exploration 3.4
Essential Question What conjectures can you make about
perpendicular lines?
1
EXPLORATION: Writing Conjectures
Work with a partner. Fold a piece of paper in half twice. Label points on the two
creases, as shown.
a. Write a conjecture about AB and CD. Justify your
D
conjecture.
A
O
B
b. Write a conjecture about AO and OB. Justify your
conjecture.
C
2
EXPLORATION: Exploring a Segment Bisector
Work with a partner. Fold and crease a piece of paper, as shown. Label the ends of the
crease as A and B.
A
a. Fold the paper again so that point A coincides with point B.
Crease the paper on that fold.
b. Unfold the paper and examine the four angles formed by
the two creases. What can you conclude about the four
angles?
Copyright © Big Ideas Learning, LLC
All rights reserved.
B
Geometry
Student Journal
81
Name _________________________________________________________ Date _________
3.4
3
Proofs with Perpendicular Lines (continued)
EXPLORATION: Writing a Conjecture
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner.
a. Draw AB, as shown.
b. Draw an arc with center A on each side of AB. Using the same compass setting,
draw an arc with center B on each side of AB. Label the intersections of the arcs
C and D.
c. Draw CD. Label its intersection with AB as O.
Write a conjecture about the resulting diagram.
Justify your conjecture.
A
O
C
D
B
Communicate Your Answer
4. What conjectures can you make about perpendicular lines?
5. In Exploration 3, find AO and OB when AB = 4 units.
82
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.4
Date __________
Notetaking with Vocabulary
For use after Lesson 3.4
In your own words, write the meaning of each vocabulary term.
distance from a point to a line
perpendicular bisector
Theorems
Theorem 3.10 Linear Pair Perpendicular Theorem
g
If two lines intersect to form a linear pair of congruent angles, then the lines
are perpendicular.
If ∠1 ≅ ∠2, then g ⊥ h.
1
2
h
Notes:
Theorem 3.11 Perpendicular Transversal Theorem
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is
perpendicular to the other line.
j
h
If h  k and j ⊥ h, then j ⊥ k .
k
Notes:
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
83
Name _________________________________________________________ Date _________
3.4
Notetaking with Vocabulary (continued)
Theorem 3.12 Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the same line, then they are
parallel to each other.
m
n
p
If m ⊥ p and n ⊥ p, then m  n.
Notes:
Extra Practice

In Exercises 1–4, find the distance from point A to BC.
1.
C
2
(−3, 0)
A
y
4
2.
y
2
(3, 2)
C(1, 4)
D(3, 2)
(4, 1)
x
B
2
4 x
−2
4
D(−1, −2)
B
−4
A
−4
(−3, −4)
(−3, −4)
3.
4
4.
y
4
B(−1, 3)
2
D(0, 1)
−4
A(−3, −3)
84
2
4 x
C(1, −1)
−4
Geometry
Student Journal
−4
−2
y
A (1, 3)
C
(3, 2)
4 x
−2
B(1, −2)
D(0, −4)
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.4
Date __________
Notetaking with Vocabulary (continued)
In Exercises 5–8, determine which lines, if any, must be parallel. Explain your
reasoning.
5.
p
q
s
6.
p
s
q
r
7.
m
n
j
k
8.
m
1
n
2
s
t
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
85
Name _________________________________________________________ Date _________
Slopes of Lines
3.5
For use with Exploration 3.5
Essential Question How can you use the slope of a line to describe
the line?
y
Slope is the rate of change between any two points on a line.
It is the measure of the steepness of the line.
6
To find the slope of a line, find the ratio of the change in y
(vertical change) to the change in x (horizontal change).
slope =
1
3
4
2
2
change in y
change in x
2
slope =
4
6
3
2
x
EXPLORATION: Finding the Slope of a Line
Work with a partner. Find the slope of each line using two methods.
Method 1: Use the two black points.
Method 2: Use the two gray points.
Do you get the same slope using each method? Why do you think this happens?
a.
2
−4
−2
2
−2
−4
86
Geometry
Student Journal
b.
y
4
y
2
4 x
−4
−2
2
4 x
−2
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.5
2
Date __________
Slopes of Lines (continued)
EXPLORATION: Drawing Lines with Given Slopes
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner.
a. Draw a line through point P using a slope of 3 .
4
4
Use the same slope to draw a line through point Q.
b. Draw a line through point R using a slope of − 4 .
3
c. What do you notice about the lines through the
points P and Q?
2
y
R
Q
−4
−2
2
4 x
P
−4
d. Describe the angle formed by the lines through points P and R. What do you
notice about the product of the slopes of the two lines?
Communicate Your Answer
3. How can you use the slope of a line to describe the line?
4. Make a conjecture about two different nonvertical lines in the same plane that
have the same slope.
5. Make a conjecture about two lines in the same plane whose slopes have a product
of −1.
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
87
Name _________________________________________________________ Date _________
3.5
Notetaking with Vocabulary
For use after Lesson 3.5
In your own words, write the meaning of each vocabulary term.
slope
directed line segment
Notes:
Core Concepts
Slopes of Lines in the Coordinate Plane
Negative slope: falls from left to right, as in line j
j
n
y
k
Positive slope: rises from left to right, as in line k
Zero slope (slope of 0): horizontal, as in line 
x
Undefined slope: vertical, as in line n
Notes:
88
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.5
Date __________
Notetaking with Vocabulary (continued)
Theorems
Theorem 3.13 Slopes of Parallel Lines
In a coordinate plane, two distinct nonvertical lines are parallel if and
only if they have the same slope.
y
x
Any two vertical lines are parallel.
Notes:
m1 = m 2
Theorem 3.14 Slopes of Perpendicular Lines
In a coordinate plane, two nonvertical lines are perpendicular if and
only if the product of their slopes is −1.
y
Horizontal lines are perpendicular to vertical lines.
x
Notes:
m1 • m 2 = −1
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
89
Name _________________________________________________________ Date _________
3.5
Notetaking with Vocabulary (continued)
Extra Practice
In Exercises 1 and 2, find the slope of the line that passes through the given
points.
1.
(4, 1), (−2, − 2)
2.
(3, 1), (6, 4)
In Exercises 3–6, find the coordinates of point P along the directed line segment
AB so that AP to PB is the given ratio.
3. A( − 2, 7), B ( − 4, 1); 3 to 1
4. A(3, 1), B (8, − 2); 2 to 3
5. Determine which of the lines are parallel and which
of the lines are perpendicular.
s
r
(−2, 4)
4
2
t (−2, 1)
−2
(−4, −1)
y
p
(1, 4)
(2, 1)
q
5 x
(1, −2)
(3, −1)
−2
(0, −4)
6. Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your
answer.
Line 1: ( 2, 0), ( − 2, 2)
Line 2: (1, − 2), ( 4, 4)
90
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
Date __________
Equations of Parallel and Perpendicular Lines
3.6
For use with Exploration 3.6
Essential Question How can you write an equation of a line that is
parallel or perpendicular to a given line and passes through a given point?
1
EXPLORATION: Writing Equations of Parallel and Perpendicular Lines
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Write an equation of the line that is parallel or perpendicular to
the given line and passes through the given point. Use a graphing calculator to verify
your answer.
a.
b.
4
4
3
(0, 2)
−6
−6
6
y=
3
x
2
(0, 1)
6
−1
−4
−4
c.
y = 2x − 1
d.
4
4
1
1
y = 2x + 2
−6
y = 2x + 2
6
−6
6
(2, −3)
(2, −2)
−4
e.
−4
f.
4
4
y = −2x + 2
y = −2x + 2
−6
6
−6
(4, 0)
(0, −2)
−4
Copyright © Big Ideas Learning, LLC
All rights reserved.
6
−4
Geometry
Student Journal
91
Name _________________________________________________________ Date _________
3.6
2
Equations of Parallel and Perpendicular Lines (continued)
EXPLORATION: Writing Equations of Parallel and Perpendicular Lines
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Write the equations of the parallel or perpendicular lines. Use
a graphing calculator to verify your answers.
a.
b.
4
−6
6
−4
4
−6
6
−4
Communicate Your Answer
3. How can you write an equation of a line that is parallel or perpendicular to a given
line and passes through a given point?
4. Write an equation of the line that is (a) parallel and (b) perpendicular to the line
y = 3x + 2 and passes through the point (1, − 2).
92
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.6
Date __________
Notetaking with Vocabulary
For use after Lesson 3.6
In your own words, write the meaning of each vocabulary term.
slope-intercept form
y-intercept
Notes:
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
93
Name _________________________________________________________ Date _________
3.6
Notetaking with Vocabulary (continued)
Extra Practice
In Exercises 1–4, write an equation of the line passing through point P that is
parallel to the given line. Graph the equations of the lines to check that they are
parallel.
1. P (3, − 2), y = 2 x − 1
3
2. P (6, 0), y = − 2( x + 2)
3. P ( − 5, − 6), x = 4
4. P ( − 4, 7), 3 x + 4 y = 8
94
Geometry
Student Journal
Copyright © Big Ideas Learning, LLC
All rights reserved.
Name_________________________________________________________
3.6
Date __________
Notetaking with Vocabulary (continued)
In Exercises 5–8, write an equation of the line passing through point P that is
perpendicular to the given line. Graph the equations of the lines to check that
they are perpendicular.
5. P ( − 2, 2), y = 2 x + 3
6. P (3, −1), x = 8
7. P ( − 4, − 3), y + 2 = − 1 ( x + 5)
2
8. P (0, 5), 4x + y = 5
Copyright © Big Ideas Learning, LLC
All rights reserved.
Geometry
Student Journal
95