5.3 Proving Triangle Congruence by SAS Essential Question

5.3
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
G.5.A
G.6.B
Proving Triangle Congruence by SAS
Essential Question
What can you conclude about two triangles
when you know that two pairs of corresponding sides and the corresponding
included angles are congruent?
Drawing Triangles
Work with a partner. Use dynamic geometry software.
a. Construct circles with radii of
2 units and 3 units centered at the
origin. Construct a 40° angle with
its vertex at the origin. Label the
vertex A.
4
3
2
1
40°
0
SELECTING TOOLS
To be proficient in math,
you need to use technology
to help visualize the results
of varying assumptions,
explore consequences,
and compare predictions
with data.
b. Locate the point where one ray
of the angle intersects the smaller
circle and label this point B.
Locate the point where the other
ray of the angle intersects the
larger circle and label this point C.
Then draw △ABC.
−4
−3
−2
−1
A
0
1
2
3
4
5
4
5
−1
−2
−3
4
3
c. Find BC, m∠B, and m∠C.
B
2
1
d. Repeat parts (a)–(c) several
times, redrawing the angle in
different positions. Keep track
of your results by copying and
completing the table below.
Write a conjecture about
your findings.
A
B
C
0
−4
−3
−2
−1
A
C
40°
0
1
2
3
−1
−2
−3
AB
AC
BC
m∠A
1.
(0, 0)
2
3
40°
2.
(0, 0)
2
3
40°
3.
(0, 0)
2
3
40°
4.
(0, 0)
2
3
40°
5.
(0, 0)
2
3
40°
m∠B
m∠C
Communicate Your Answer
2. What can you conclude about two triangles when you know that two pairs of
corresponding sides and the corresponding included angles are congruent?
3. How would you prove your conjecture in Exploration 1(d)?
Section 5.3
Proving Triangle Congruence by SAS
249
5.3 Lesson
What You Will Learn
Use the Side-Angle-Side (SAS) Congruence Theorem.
Solve real-life problems.
Core Vocabul
Vocabulary
larry
Using the Side-Angle-Side Congruence Theorem
Previous
congruent figures
rigid motion
Theorem
Theorem 5.5 Side-Angle-Side (SAS) Congruence Theorem
If two sides and the included angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two triangles are congruent.
STUDY TIP
The included angle of
two sides of a triangle is
the angle formed by the
two sides.
— ≅ DE
—, ∠A ≅ ∠D, and AC
— ≅ DF
—,
If AB
then △ ABC ≅ △DEF.
E
B
F
C
Proof p. 250
A
D
Side-Angle-Side (SAS) Congruence Theorem
— ≅ DE
—, ∠A ≅ ∠D, AC
— ≅ DF
—
Given AB
E
B
Prove △ABC ≅ △DEF
F
C
A
D
First, translate △ABC so that point A maps to point D, as shown below.
E
B
E
B′
F
C
A
F
D
C′
D
This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise
through ∠C′DF so that the image of ⃗
DC′ coincides with ⃗
DF, as shown below.
E
E
B′
F
F
D
C′
D
B″
— ≅ DF
—, the rotation maps point C′ to point F. So, this rotation maps
Because DC′
△DB′C′ to △DB″F. Now, reflect △DB″F in the line through points D and F, as
shown below.
E
E
F
F
D
B″
D
Because points D and F lie on ⃖⃗
DF, this reflection maps them onto themselves. Because
a reflection preserves angle measure and ∠B″DF ≅ ∠EDF, the reflection maps ⃗
DB″
— ≅ DE
—, the reflection maps point B″ to point E. So, this reflection
to ⃗
DE. Because DB″
maps △DB″F to △DEF.
Because you can map △ABC to △DEF using a composition of rigid motions,
△ABC ≅ △DEF.
250
Chapter 5
Congruent Triangles
Using the SAS Congruence Theorem
B
Write a proof.
STUDY TIP
Make your proof easier
to read by identifying the
steps where you show
congruent sides (S) and
angles (A).
— ≅ DA
—, BC
— AD
—
Given BC
C
Prove △ABC ≅ △CDA
A
D
SOLUTION
STATEMENTS
REASONS
S
1. Given
— —
1. BC ≅ DA
— —
2. BC AD
2. Given
A 3. ∠BCA ≅ ∠DAC
— ≅ CA
—
S 4. AC
3. Alternate Interior Angles Theorem (Thm. 3.2)
4. Reflexive Property of Congruence (Thm. 2.1)
5. △ABC ≅ △CDA
5. SAS Congruence Theorem
Using SAS and Properties of Shapes
— and RP
— pass through the center M of the circle. What can you
In the diagram, QS
conclude about △MRS and △MPQ?
S
R
P
M
Q
SOLUTION
Because they are vertical angles, ∠PMQ ≅ ∠RMS. All points on a circle are the same
—, MQ
—, MR
—, and MS
— are all congruent.
distance from the center, so MP
So, △MRS and △MPQ are congruent by the SAS Congruence Theorem.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
In the diagram, ABCD is a square with four congruent sides and four right
— ⊥ SU
—
angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT
— ≅ VU
—.
and SV
S
B
R
A
V
U
C
T
D
1. Prove that △SVR ≅ △UVR.
2. Prove that △BSR ≅ △DUT.
Section 5.3
Proving Triangle Congruence by SAS
251
Copying a Triangle Using SAS
C
Construct a triangle that is congruent to △ABC using the
SAS Congruence Theorem. Use a compass and straightedge.
A
SOLUTION
Step 1
Step 2
Step 3
Step 4
F
D
E
Construct a side
— so that it
Construct DE
—.
is congruent to AB
D
E
Construct an angle
Construct ∠D with vertex
D and side ⃗
DE so that it is
congruent to ∠A.
B
D
F
E
D
Construct a side
— so that
Construct DF
—.
it is congruent to AC
E
Draw a triangle
Draw △DEF. By the SAS
Congruence Theorem,
△ABC ≅ △DEF.
Solving Real-Life Problems
Solving a Real-Life Problem
You are making a canvas sign to hang
on the triangular portion of the barn wall
shown in the picture. You think you can
use two identical triangular sheets of
— ⊥ QS
— and
canvas. You know that RP
— ≅ PS
—. Use the SAS Congruence
PQ
Theorem to show that △PQR ≅ △PSR.
R
S
Q
P
SOLUTION
— ≅ PS
—. By the Reflexive Property of Congruence (Theorem 2.1),
You are given that PQ
—
—
RP ≅ RP . By the definition of perpendicular lines, both ∠RPQ and ∠RPS are right
angles, so they are congruent. So, two pairs of sides and their included angles are
congruent.
△PQR and △PSR are congruent by the SAS Congruence Theorem.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
3. You are designing the window shown in the photo. You want to make △DRA
— ≅ DG
— and
congruent to △DRG. You design the window so that DA
∠ADR ≅ ∠GDR. Use the SAS Congruence Theorem to prove △DRA ≅ △DRG.
D
A
252
Chapter 5
Congruent Triangles
R
G
Exercises
5.3
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. WRITING What is an included angle?
2. COMPLETE THE SENTENCE If two sides and the included angle of one triangle are congruent to
two sides and the included angle of a second triangle, then ___________.
Monitoring Progress and Modeling with Mathematics
In Exercises 15–18, write a proof. (See Example 1.)
In Exercises 3–8, name the included angle between the
pair of sides given.
J
—
— —
15. Given PQ bisects ∠SPT, SP ≅ TP
L
Prove △SPQ ≅ △TPQ
P
K
P
S
— and KL
—
3. JK
— and LK
—
4. PK
5.
— and LK
—
LP
6.
— and JK
—
JL
7.
— and JL
—
KL
8.
— and PL
—
KP
Q
— ≅ CD
—, AB
— CD
—
16. Given AB
Prove △ABC ≅ △CDA
A
In Exercises 9–14, decide whether enough information
is given to prove that the triangles are congruent using
the SAS Congruence Theorem (Theorem 5.5). Explain.
9. △ABD, △CDB
A
1
2
M
L
B
Q
N
B
C
11. △YXZ, △WXZ
Prove △ABC ≅ △EDC
D
12. △QRV, △TSU
R
S
A
Q
13. △EFH, △GHF
V
U
K
B
T
—≅ RT
—, QT
— ≅ ST
—
18. Given PT
14. △KLM, △MNK
E
E
C
Y
F
C
— and BD
—.
17. Given C is the midpoint of AE
P
Z
X
D
10. △LMN, △NQP
D
W
T
L
Prove △PQT ≅ △RST
P
Q
T
G
H
N
M
S
Section 5.3
R
Proving Triangle Congruence by SAS
253
27. PROOF The Navajo rug is made of isosceles
In Exercises 19–22, use the given information to
name two triangles that are congruent. Explain your
reasoning. (See Example 2.)
19. ∠SRT ≅ ∠URT, and
20. ABCD is a square with
R is the center of
the circle.
S
four congruent sides and
four congruent angles.
B
D
C
B
T
R
triangles. You know ∠B ≅ ∠D. Use the SAS
Congruence Theorem (Theorem 5.5) to show that
△ABC ≅ △CDE. (See Example 3.)
A
C
E
U
A
D
— ⊥ MN
—, KL
— ⊥ NL
—,
22. MK
21. RSTUV is a regular
28. HOW DO YOU SEE IT?
and M and L are centers
of circles.
pentagon.
T
B
What additional information
do you need to prove that
△ABC ≅ △DBC?
K
S
U
10 m
M
R
A
L
10 m
V
△ABC ≅ △DEC.
Then find the values
of x and y.
CONSTRUCTION In Exercises 23 and 24, construct a
triangle that is congruent to △ABC using the SAS
Congruence Theorem (Theorem 5.5).
A
4y − 6
3y + 1
D
2x + 6
C
B
24. B
B
D
29. MATHEMATICAL CONNECTIONS Prove that
N
23.
C
4x
E
30. THOUGHT PROVOKING There are six possible subsets
A
A
C
C
of three sides or angles of a triangle: SSS, SAS, SSA,
AAA, ASA, and AAS. Which of these correspond to
congruence theorems? For those that do not, give a
counterexample.
25. ERROR ANALYSIS Describe and correct the error in
finding the value of x.
✗
Y
5x − 1
4x + 6
X
W
5x − 5 Z
3x + 9
31. MAKING AN ARGUMENT Your friend claims it is
4x + 6 = 3x + 9
x+6=9
x=3
possible to construct a triangle
congruent to △ABC by first
— and AC
—, and
constructing AB
then copying ∠C. Is your
friend correct? Explain
your reasoning.
26. WRITING Describe the relationship between your
conjecture in Exploration 1(d) on page 249 and the
Side-Angle-Side (SAS) Congruence Theorem
(Thm. 5.5).
Intersecting Lines Theorem (Theorem 4.3).
Reviewing what you learned in previous grades and lessons
Classify the triangle by its sides and by measuring its angles. (Section 5.1)
254
34.
Chapter 5
Congruent Triangles
A
32. PROVING A THEOREM Prove the Reflections in
Maintaining Mathematical Proficiency
33.
C
35.
36.
B