Download Lines and Angles

MODULE
STUDY GUIDE REVIEW
19
Lines and Angles
Essential Question: How can you use parallel and perpendicular
lines to solve real-world problems?
KEY EXAMPLE
(Lesson 19.1)
Find m∠ABD given that m∠CBE = 40° and the angles are formed
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by the intersection of the lines AC and DE at point B.
When two lines intersect, they form two pairs of vertical angles at
their intersection. Note that ∠ABD and ∠CBE are vertical angles and
∠DBC and ∠ABE are vertical angles.
∠ABD ≅ ∠CBE
m∠ABD = m∠CBE = 40°
Key Vocabulary
vertical angles
(ángulos verticales)
complementary angles
(ángulos complementarios)
supplementary angles
(ángulos suplementarios)
transversal (transversal)
indirect proof (prueba
indirecta)
Vertical Angles Theorem
Definition of congruence of angles
KEY EXAMPLE
(Lesson 19.2)
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Find m∠APD given that AB intersects the parallel lines DE and FG at the points
P and Q, respectively, and m∠AQF = 70°.
When a transversal intersects two parallel lines, it forms a series of angle pairs. Note that ∠APD and
∠AQF are a pair of corresponding angles.
m∠APD = m∠AQF
m∠APD = 70°
Corresponding Angles Theorem
Substitute the known angle measure.
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KEY EXAMPLE
(Lesson 19.3)
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Determine whether the lines DE and FG are parallel given that AB intersects
them at the points P and Q, respectively, m∠APE = 60°, and m∠BQF = 60°.
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Lines AB and DE intersect, so they create two pairs of vertical angles. The angle which is the opposite
of ∠APE is ∠DPB, so they are called vertical angles.
∠APE ≅ ∠DPB
m∠APE = m∠DPB
m∠DPB = 60°
Vertical Angles Theorem
Definition of congruence
Substitute the known angle measure.
m∠BQF = m∠DPB = 60°
∠BQF ≅ ∠DPB
Definition of congruence
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Thus, the lines DE and FG are parallel by the converse of the Corresponding Angles Theorem because
their corresponding angles are congruent.
Module 19
983
Study Guide Review
EXERCISES
Find the angle measure.
1.
m∠ABD given that m∠CBD = 40° and the
angles are formed by the intersection of the
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lines AC and DE at point B. (Lesson 19.1)
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2. m∠BPE given that AB intersects the parallel
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lines DE and FG at the points P and Q,
respectively, and m∠AQF = 45°. (Lesson 19.2)
Determine whether the lines are parallel. (Lesson 19.3)
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3. DE and FG , given that AB intersects them at the points P and Q, respectively, m∠APD = 60°, and
m∠BQG = 120°.
Find the distance and angle formed from the perpendicular bisector. (Lesson 19.4)
4. Find the distance
of point D from
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_B given that D is the point at the perpendicular bisector of the
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line segment AB, DE intersects AB, and AD = 3. Find m∠ADE.
Find the equation of the line. (Lesson 19.5)
2 x + 2 and passes through the point (3, 4).
5. Perpendicular to y = __
3
MODULE PERFORMANCE TASK
Mystery Spot Geometry
Inside mystery spot buildings, some odd things can appear to occur. Water can appear
to flow uphill, and people can look as if they are standing at impossible angles. That is
because there is no view of the outside, so the room appears to be normal.
The illustration shows a mystery spot building constructed
so that the floor is at a 25° angle with the ground.
View from inside
View from
inside
Use your own paper to complete the task. Use sketches, words, or geometry to explain
how you reached your conclusions.
Module 19
984
Study Guide Review
© Houghton Mifflin Harcourt Publishing Company
• A table is placed in the room with its legs
perpendicular to the floor and the tabletop
perpendicular to the legs. Sketch or describe the
View from outside
relationship of the tabletop to the floor, walls, and
View from outside
ceiling of the room. What would happen if a ball
were placed on the table?
• A chandelier hangs from the ceiling of the room. How does
it appear to someone inside? How does it appear to someone
standing outside of the room?
Ready to Go On?
19.1–19.5 Lines and Angles
• Online Homework
• Hints and Help
• Extra Practice
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Find the measure of each angle. Assume lines GB and FC are parallel.
(Lessons 19.1, 19.2)
1. The measure of ∠WOX is 70°. Find m∠YOZ.
B
A
X
2. The measure of ∠AXB is 40°. Find m∠FZE.
H
C
W
O
Y
3. The measure of ∠XWO is 70°. Find m∠OYC.
G
4. The measure of ∠BXO is 110°. Find m∠OZF.
D
Z
F
E
PB is the perpendicular bisector of ¯
AC . ¯
QC is the
Use the diagram to find lengths. ¯
¯
perpendicular bisector of BD. AB = BC = CD. (Lesson 19.4)
Q
5. Given BD = 24 and PC = 13, find PB.
P
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6. Given QB = 23 and BC = 12, find QD.
A
Find the equation of each line. (Lessons 19.5)
B
C
D
3 x + 5 and passing through the point (-7, -1)
7. The line parallel to y = -__
7
1 x + 3 and passing through the point (2, 7)
8. The line perpendicular to y = __
5
9. The perpendicular bisector to the line segment between (-3, 8) and (9, 4)
ESSENTIAL QUESTION
10. Say you want to create a ladder. Which lines should be parallel or perpendicular to each other?
Module 19
985
Study Guide Review
MODULE 19
MIXED REVIEW
Assessment Readiness
1. Consider each equation. Is it the equation of a line that is parallel or perpendicular
to y = 3x + 2?
Select Yes or No for A–C.
1x - 8
A. y = -_
Yes
No
3
B. y = 3x - 10
Yes
No
C. y = 2x + 4
Yes
No
2. Consider the following statements about △ABC. Choose True or False for each
statement.
C
A. AC = BC
True
False
B. CD = BC
True
False
C. AD = BD
True
False
A
D
B
3. The measure of angle 3 is 130° and the measure of angle 4 is 50°. State two different
relationships that can be used to prove m∠1 = 130°.
4
1
3
2
4. m∠1 = 110° and m∠6 = 70°. Use angle
relationships to show that lines m and n are parallel.
5 6
7 8
m
n
ℓ
Module 19
986
Study Guide Review
© Houghton Mifflin Harcourt Publishing Company
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