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Transcript
LESSON
5.2
Name
ASA Triangle
Congruence
5.2
Explore 1
G.6.B
Mathematical Processes
G.1.E
Create and use representations to organize, record, and communicate
mathematical ideas.
Drawing Triangles Given Two Angles
and a Side

Draw a segment that is 4 inches long. Label the endpoints A and B.

_
Use a protractor to draw a 30° angle so that one side is AB and
its vertex is point A.
1.A, 1.B, 1.F, 2.C.4, 3.H, 4.D

© Houghton Mifflin Harcourt Publishing Company
Have students work in pairs to label and color code congruent angles and
a side in pairs of triangles.
C
_
Use a protractor to draw a 40° angle so that one side is AB and
its vertex is point B. Label the point where the sides of the
angles intersect as point C.

Language Objective
A
1.
In a polygon, the side that connects two consecutive angles is the included side of
those two angles. Describe the triangle you drew using the term included side. Be as
precise as possible.
It is a triangle with a 30° angle, a 40° angle, and an included side that is
4 inches long.
2.
Discussion Based on your results, how can you decide whether two triangles
are congruent without checking that all six pairs of corresponding sides and
corresponding angles are congruent?
Possible answer: If two angles and the included side of one triangle are
40°
B
congruent to two angles and the included side of another triangle, then
the triangles are congruent.
Module 5
ges
EDIT--Chan
DO NOT Key=TX-A
Correction
must be
Lesson 2
261
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G.6.B Prove
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Also G.3.B,
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5.2
Resource
Locker
Quest
Essential
HARDCOVER PAGES 217226
Tria
Drawing
and a Side
ore 1
Expl
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of
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B.
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Draw a segme

4 inches long.
a
ctor to draw
Use a protra
is point A.
its vertex


30° angle
Label
so that one
_
and
side is AB
Turn to these pages to
find this lesson in the
hardcover student
edition.
C
_
and
side is AB
A
so that one
of the
a 40° angle
the sides
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point where
Use a protra
Label the
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C.
its vertex
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Is there a
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What does
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Reflect
© Houghto
n Mifflin
Harcour t
Publishin
1.
Lesson 2
261
Module 5
5L2 261
86_U2M0
ESE3538
GE_MTX
Lesson 5.2
4 in.
Reflect
GE_MTXESE353886_U2M05L2 261
261
30°
Put your triangle and a classmate’s triangle beside each other. Is there a
sequence of rigid motions that maps one to the other? What does this
tell you about the triangles?
Yes; the triangles are congruent.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Explain that the flags
of many countries incorporate geometric objects such
as triangles. Then preview the Lesson
Performance Task.
Resource
Locker
You have seen that two triangles are congruent if they have six pairs of congruent
corresponding parts. However, it is not always possible to check all three pairs of corresponding
sides and all three pairs of corresponding angles. Fortunately, there are shortcuts for
determining whether two triangles are congruent.
Prove two triangles are congruent by applying the Side-Angle-Side,
Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions. Also G.3.B, G.5.A, G.5.C
If two angles and the included side of one triangle
are congruent to two angles and the included side
of another triangle, the triangles are congruent.
ASA Triangle Congruence
G.6.B Prove two triangles are congruent by applying the ... Angle-Side-Angle ... congruence
conditions. Also G.3.B, G.5.A, G.5.C
The student is expected to:
Essential Question: What does the
ASA Triangle Congruence Theorem tell
you about triangles?
Date
Essential Question: What does the ASA Triangle Congruence Theorem tell you about
triangles?
Texas Math Standards
ENGAGE
Class
2/21/14
4:49 AM
2/21/14 4:49 AM
Explore 2
Justifying ASA Triangle Congruence
EXPLORE 1
Explain the results of Explore 1 using transformations.
A
Use tracing paper to make two copies of the triangle from Explore 1 as shown. Identify
the corresponding parts you know to be congruent and mark these congruent parts on
the figure.
F
∠D
∠A ≅
A
D
∠E
E
∠B ≅
B
_
_
DE
AB ≅
C
Drawing Triangles Given Two Angles
and a Side
INTEGRATE TECHNOLOGY
B
What can you do to show that these triangles are congruent?
Find a sequence of rigid motions that maps one triangle onto the other triangle.
C
translate △ABC so that point A maps to point D. What translation vector did you use?
⇀
the vector with initial point A and terminal point D AD
D
E
F
G
( )
D
A
Use a rotation to map point B to point E. What is the center of the rotation?
What is the angle of the rotation?
The center of the rotation is point D (or A); the angle of the rotation is m∠EDB.
How do you know_
the image
_ of point B is point E?
It is given that AB ≈ DE, so the image of point B must be point E.
What rigid motion do you think will map point C to point F ?
‹ ›
−
reflection across DE
QUESTIONING STRATEGIES
F
B
C
F
D
A
E
B
C
EXPLORE 2
‹ ›
−
to show that the image of point C is point F, notice that ∠A is reflected across DE, so
the measure of the angle is preserved. Since ∠A ≅ ∠D you can conclude that the image
→
→
_
‾
‾
of AC lies on DF
. In particular, the image of point C must lie on DF
. By similar
→
→
_
‾
‾
reasoning, the image of BC lies on EF
and the image of point C must lie on EF
.
_
_
the only point that lies on both DF and EF is point F .
Describe the sequence of rigid motions used to map △ABC to △DEF.
a translation followed by a rotation followed by a reflection
Reflect
3.
Discussion Arturo said the argument in the activity works for any triangles with two pairs of congruent
corresponding angles, and it is not necessary for the included sides to be congruent. Do you agree? Explain.
No; the included sides must be congruent to conclude that the image of
point B is point E after a rotation around point D.
Module 5
262
How can you check whether the triangles you
draw are congruent to the triangles your
classmates draw? Place one student’s page on top of
the other student’s page and check to see if the
triangles can be made to coincide exactly.
E
Justifying ASA Triangle Congruence
© Houghton Mifflin Harcourt Publishing Company
H
Have students explore the Angle-Side-Angle theorem
using geometry software.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Have students respond to this prompt in their math
journals.
“If I know that two pairs of corresponding angles and
the included sides of two triangles are congruent, I
know ________. I know this because ________.”
QUESTIONING STRATEGIES
Lesson 2
PROFESSIONAL DEVELOPMENT
GE_MTXESE353886_U2M05L2 262
Math Background
Students know that when triangles are congruent, all pairs of corresponding sides
and corresponding angles are congruent. As an extension of that, students to
begin to develop converses of the statement that Corresponding Parts of
Congruent Triangles Are Congruent in which they do not need to know that all
six pairs of corresponding parts are congruent in order to prove that triangles are
congruent. In this lesson, they explore the Angle-Side-Angle (ASA) Theorem.
They find that if they can prove that two angles and the included side of one
triangle are congruent to two angles and the included side of another triangle, the
triangles are congruent.
1/22/15 3:04 AM
What is the benefit of using the Angle-SideAngle Theorem instead of CPCTC? You need
to find only three pairs of congruent corresponding
parts with ASA, as opposed to six pairs with CPCTC.
ASA Triangle Congruence 262
Explain 1
EXPLAIN 1
You can state your findings about triangle congruence as a theorem. This theorem can help you
decide whether two triangles are congruent.
Deciding Whether Triangles Are
Congruent Using ASA Triangle
Congruence
ASA Triangle Congruence Theorem
If two angles and the included side of one triangle are congruent to two angles and
the included side of another triangle, then the triangles are congruent.
Example 1
CONNECT VOCABULARY
Remind students that the ASA Triangle Congruence
Theorem is a shortened form of its full name, the
Angle-Side-Angle Triangle Congruence Theorem.
When you write ASA, it is helpful to read it aloud as
Angle-Side-Angle and have students do the same to
reinforce what it means.

Give each pair pictures of congruent and
non-congruent triangles, highlighters, protractors,
and rulers. Instruct them to prove which pairs are
congruent by using Angle-Side-Angle to prove it.
Have students highlight the angles and the side they
used to show congruence, and write notes explaining
why the triangles are congruent.
E
B
74°
m∠D + m∠E + m∠F = 180°
m∠D + 74° + 61° = 180°
m∠D + 135° = 180°
A
61°
45°
2.3 cm
m∠D = 45°
D
C
61°
2.3 cm
F
Step 2 Compare the angle measures and side lengths.
m∠A = m∠D = 45°, AC = DF = 2.3 cm, and m∠C = m∠F = 61°
_ _
So, ∠A ≅ ∠D, AC ≅ DF, and ∠C ≅ ∠F.
_
_
∠A and ∠C include side AC, and ∠D and ∠F include side DF.
So, △ABC ≅ △DEF by the ASA Triangle Congruence Theorem.

J
Step 1 Find m∠P.
m∠M + m∠N + m∠P = 180°
69 ° + m∠P = 180°
m∠P =
111
31°
K
62 in.
31 ° + 38 ° + m∠P = 180°
© Houghton Mifflin Harcourt Publishing Company
LANGUAGE SUPPORT
Determine whether the triangles are congruent. Explain
your reasoning.
Step 1 Find m∠D.
QUESTIONING STRATEGIES
Why do you need to find the measure of the
missing angle to use the ASA Triangle
Congruence Theorem? The two sides that you know
are congruent need to be the included sides, so you
need to know the measures of the angles at their
endpoints.
Deciding Whether Triangles Are Congruent
Using ASA Triangle Congruence
110°
L
P
°
Step 2 Compare the angle measures and side lengths.
62 in.
N
38°
31°
M
None of the angles in △MNP has a measure of 110° .
Therefore, there is/is not a sequence of rigid motions that maps
△MNP onto △JKL, and △MNP is/is not congruent to △JKL.
Reflect
4.
In Part B, do you need to find m∠K? Why or why not?
No; you only need to know that △JKL has an angle (∠L) that is not congruent to any
angle of △MNP. At that point, you can conclude that the triangles are not congruent.
Module 5
263
Lesson 2
COLLABORATIVE LEARNING
GE_MTXESE353886_U2M05L2 263
Small Group Activity
Have students experiment with congruent triangles and triangles that are not
congruent but do have some congruent parts. Instruct them to draw a pair of
congruent triangles and a pair of non-congruent triangles that meet the following
criteria:
• at least two pairs of congruent sides
• at least one pair each of congruent sides and congruent angles
• all three pairs of congruent angles
• at least two pairs of congruent sides and one pair of congruent angles
263
Lesson 5.2
24/02/14 6:08 AM
Your Turn
EXPLAIN 2
Determine whether the triangles are congruent. Explain your reasoning.
5.
6.
A
P
Q
B
D
C
_ _
∠B ≅ ∠C, BD ≅ CD, and ∠ADB ≅
∠ADC since both are right angles.
_
BD and
∠B and ∠ADB include side_
∠ADC and ∠C include side DC. So,
△ADB ≅ △ADC by the ASA Triangle
Congruence Theorem.
Explain 2
72°
38°
1 in.
S
R
Proving Triangles Are Congruent
Using ASA Triangle Congruence
38° 67° U
1 in.
T
72° + 38° + m∠R = 180°
AVOID COMMON ERRORS
None of the angles in △PQR has a
measure of 67°. So, △PQR is not
congruent to △STU.
Some students may forget to include in their proofs
the information that is given in the diagram. Remind
them to start the proof by listing the given
information.
m∠R = 70°
Proving Triangles Are Congruent
Using ASA Triangle Congruence
QUESTIONING STRATEGIES
The ASA Triangle Congruence Theorem may be used as a reason in a proof.
Example 2

Write each proof.
How do you know when you have enough
information to complete the proof? To
complete the proof, you need to show that two
angles and the included side of one triangle are
congruent to the corresponding angles and side of
the other triangle.
M
Given: ∠MQP ≅ ∠NPQ, ∠MPQ ≅ ∠NQP
Prove: △MQP ≅ △NPQ
Q
P
N
Reasons
1. ∠MQP ≅ ∠NPQ
1. Given
2. ∠MPQ ≅ ∠NQP
2. Given
_ _
3. QP ≅ QP
3. Reflexive Property of Congruence
4. △MQP ≅ △NPQ
4. ASA Triangle Congruence Theorem
Module 5
264
CONNECT VOCABULARY
© Houghton Mifflin Harcourt Publishing Company
Statements
Have students review the different units used to
measure triangles to show congruence. Length is
measured in linear units such as cm, mm, or inches,
while angles are measured in degrees.
Lesson 2
DIFFERENTIATE INSTRUCTION
GE_MTXESE353886_U2M05L2 264
Modeling
1/22/15 3:04 AM
Instruct students to draw and label three triangles according to the following
specifications.
• There is just enough information to prove that they are congruent using the
ASA Triangle Congruence Theorem.
• There is enough information to prove that they are not congruent.
• There is some of the information required to prove that they are congruent,
but not enough.
ASA Triangle Congruence 264
D
_
Given: ∠A ≅ ∠C, E is the midpoint of AC.
B
A
Prove: △AEB ≅ △CED
E
C
B
Statements
1. ∠A ≅ ∠C
Reasons
1.
Given
_
2. E is the midpoint of AC .
2.
Given
_ _
3. AE ≅ CE
3.
Definition of midpoint
4. ∠AEB ≅ ∠CED
4.
Vertical angles are congruent.
5. △AEB ≅ △CED
5.
ASA Triangle Congruence Theorem
Reflect
7.
_
In Part B, suppose the length of AB is 8.2 centimeters. Can you determine the length
of any other segments in the figure?
_
_ Explain.
Yes; CD = 8.2 cm because AB ≅ CD by CPCTC.
Your Turn
Write each proof.
8.
Given: ∠JLM ≅ ∠KML, ∠JML ≅ ∠KLM
J
Prove: △JML ≅ △KLM
K
© Houghton Mifflin Harcourt Publishing Company
L
Statements
M
Reasons
1. ∠JLM ≅ ∠KML
1. Given
2. ∠JML ≅ ∠KLM
_ _
3. LM ≅ LM
2. Given
4. △JML ≅ △KLM
4. ASA Triangle Congruence Theorem
Module 5
3. Reflexive Property of Congruence
265
Lesson 2
LANGUAGE SUPPORT
GE_MTXESE353886_U2M05L2 265
Vocabulary Development
Make sure students understand the meaning of included side. Sketch 4BER,
_
4MAT, and 4SQU on the board. Use color to highlight SQ and define what it
means for it to be included between ∠S and ∠Q. Ask questions such as “What
_
side is included between ∠E and ∠R?” and “Between which two angles is MT ?”
Continue to drill students until they can recognize and name included sides
fluently.
265
Lesson 5.2
1/22/15 3:04 AM
9.
_
_
Given: ∠S and ∠U are right angles, RV bisects SU.
R
ELABORATE
Prove: △RST ≅ △VUT
S
T
U
VISUAL CUES
Use colored pencils to label congruent sides using
ticks and congruent angles using arcs to help students
better visualize the angles and sides that are
congruent
V
Statements
Reasons
1. ∠S and ∠U are right angles.
1. Given
2. ∠S ≅ ∠U
_
_
3. RV bisects SU.
_ _
4. ST ≅ UT
2. All right angles are congruent.
5. ∠RTS ≅ ∠VTU
5. Vertical angles are congruent.
6. △RST ≅ △VUT
6. ASA Triangle Congruence Theorem
3. Given
SUMMARIZE THE LESSON
4. Definition of bisector
Why would you use the ASA Triangle
Congruence Theorem? What do you need to
know to use it? You would use the ASA Triangle
Congruence Theorem to prove that two triangles
are congruent by using only three pairs of
congruent parts. You need to know that two pairs of
corresponding angles are congruent and the
included sides between those angles are also
congruent.
Elaborate
10. Discussion Suppose you and a classmate both draw triangles with a 30° angle, a
70° angle, and a side that is 3 inches long. How will they compare? Explain your
reasoning.
The triangles will be congruent by the ASA Triangle Congruence Theorem
if the 3 inch side is the included side. Otherwise, the triangles will have
the same shape but not necessarily the same size.
© Houghton Mifflin Harcourt Publishing Company
11. Discussion How can a diagram show you that corresponding parts of two triangles
are congruent without providing specific angle measures or side lengths?
Possible answer: Vertical angles are congruent. Overlapping sides are
congruent. Right angles are congruent. Angles or sides marked with
congruence symbols are congruent.
12. Essential Question Check-In What must be true in order for you to use the ASA
Triangle Congruence Theorem to prove that triangles are congruent?
Two angles and the included side of one triangle must be congruent to two
angles and the included side of another triangle.
Module 5
GE_MTXESE353886_U2M05L2 266
266
Lesson 2
2/21/14 4:49 AM
ASA Triangle Congruence 266