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Topic 4
Congruent Triangles
TOPIC OVERVIEW
VOCABULARY
4-1
Congruent Figures
4-2
Triangle Congruence by SSS and SAS
4-3
Triangle Congruence by ASA
and AAS
4-4
Using Corresponding Parts of
Congruent Triangles
4-5
Isosceles and Equilateral Triangles
4-6
Congruence in Right Triangles
4-7
Congruence in Overlapping Triangles
DIGITAL
APPS
English/Spanish Vocabulary Audio Online:
EnglishSpanish
base of an isosceles triangle, p. 168
base de un triángulo isósceles
base angles of an isosceles
ángulos de la base de un
triangle, p. 168
triángulo isósceles
congruent polygons, p. 148
polígonos congruentes
corollary, p. 168corolario
hypotenuse, p. 174hipotenusa
legs of an isosceles triangle, p. 168
catetos de un triángulo isósceles
legs of a right triangle, p. 174
catetos de un triángulo rectángulo
vertex angle of an isosceles
ángulo del vértice de un triángulo
triangle, p. 174isósceles
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Topic 4 Congruent Triangles
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4-1 Congruent Figures
TEKS FOCUS
VOCABULARY
TEKS (6)(C) Apply the definition of
congruence, in terms of rigid transformations,
to identify congruent figures and their
corresponding sides and angles.
TEKS (1)(B) Use a problem-solving
model that incorporates analyzing given
information, formulating a plan or strategy,
determining a solution, justifying the solution,
and evaluating the problem-solving process
and the reasonableness of the solution.
•Congruent polygons – polygons that have congruent corresponding
sides and angles
•Formulate – create with careful effort and purpose. You can
formulate a plan or strategy to solve a problem.
•Reasonableness – the quality of being within the realm of common
sense or sound reasoning. The reasonableness of a solution is whether
or not the solution makes sense.
•Strategy – a plan or method for solving a problem
Additional TEKS (1)(F), (1)(G)
ESSENTIAL UNDERSTANDING
You can determine whether two figures are congruent by comparing their corresponding
sides and angles.
Key Concept Congruent Figures
Definition
Example
Congruent polygons have
congruent corresponding sides
and angles. When you name
congruent polygons, you must
list corresponding vertices in the
same order.
A
B
F
E
D
C
G
H
AB ≅ EF BC ≅ FG
CD ≅ GH DA ≅ HE
∠A ≅ ∠E ∠B ≅ ∠F
∠C ≅ ∠G ∠D ≅ ∠H
ABCD ≅ EFGH
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Theorem 4-1 Third Angles Theorem
Theorem
If two angles of one triangle
are congruent to two angles of
another triangle, then the third
angles are congruent.
If . . .
∠A ≅ ∠D and ∠B ≅ ∠E
D
A
B
Then . . .
∠C ≅ ∠F
C
E
F
You will prove Theorem 4-1 in Exercise 37.
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148
Lesson 4-1 Congruent Figures
Problem 1
Finding Congruent Sides and Angles
How do you know
which sides and
angles correspond?
The congruence statement
HIJK ≅ LMNO tells you
which parts correspond.
K
If HIJK ≅ LMNO, what are the congruent corresponding parts?
Sides:
HI ≅ LM
Angles: ∠H ≅ ∠L
IJ ≅ MN JK ≅ NO
KH ≅ OL
∠I ≅ ∠M
∠J ≅ ∠N ∠K ≅ ∠O
L
J O
H
N
I
Problem 2
M
TEKS Process Standard (1)(B)
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Using Congruent Sides and Angles
You know two angle
measures in △ABC.
How can they help?
In the congruent triangles, ∠D corresponds to
∠A, so you know that
∠D ≅ ∠A. You can find
m∠D by first finding
m∠A.
Multiple Choice The wings of an SR-71 Blackbird aircraft suggest
congruent triangles. What is mjD?
30
105
75
150
Use the Triangle AngleSum Theorem to write an
equation involving m∠A.
Solve for m∠A.
∠A and ∠D are
corresponding angles of
congruent triangles, so
∠A ≅ ∠D.
m∠A + 30 + 75 = 180
m∠A + 105 = 180
m∠A = 75
m∠A = m∠D = 75
The correct answer
is B.
Problem 3
TEKS Process Standard (1)(G)
Finding Congruent Triangles
How do you
determine whether
two triangles are
congruent?
Compare each pair of
corresponding parts. If all
six pairs are congruent,
then the triangles are
congruent.
Are the triangles congruent? Justify your answer.
AB ≅ ED
Given
BC ≅ DC
BC = 4 = DC
AC ≅ EC
AC = 6 = EC
∠A ≅ ∠E, ∠B ≅ ∠D
Given
∠BCA ≅ ∠DCE
Vertical angles are congruent.
△ABC ≅ △EDC by the definition of congruent triangles.
B
A
4
6
E
6
C 4
D
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149
Problem 4
Proof Proving Triangles Congruent
Given: LM ≅ LO, MN ≅ ON,
∠M ≅ ∠O, ∠MLN ≅ ∠OLN
HO
ME
RK
O
You know four pairs
of congruent parts.
What else do you
need to prove the
triangles congruent?
You need a third pair of
congruent sides and a
third pair of congruent
angles.
NLINE
WO
M
L
N
Prove: △LMN ≅ △LON
O
Statements
Reasons
1) LM ≅ LO, MN ≅ ON
1) Given
2) LN ≅ LN
2) Reflexive Property of ≅
3) ∠M ≅ ∠O, ∠MLN ≅ ∠OLN
3) Given
4) ∠MNL ≅ ∠ONL
4) Third Angles Theorem
5) △LMN ≅ △LON
5) Definition of ≅ triangles
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PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Scan page for a Virtual Nerd™ tutorial video.
1.
Apply Mathematics (1)(A) Builders use the king post truss (below left) for
the top of a simple structure. In this truss, △ABC ≅ △ABD. List the congruent
corresponding sides and angles.
A
H
E
G
J
C
D
B
I
F
king
post
truss
attic
frame
truss
2.
The attic frame truss (above right) provides open space in the center for storage.
In this truss, △EFG ≅ △HIJ. List the congruent corresponding sides and angles.
△LMC ≅ △BJK. Complete the congruence statements.
J
M
3.
LC ≅ ? 4.KJ ≅ ?
hsm11gmse_0401_t02427
5.
∠K ≅ ? 7.
△CML ≅ ? 6.∠M ≅ ?
8.△KBJ ≅ ?
L
C K
B
POLY @ SIDE. List each of the following.
9.
four pairs of congruent sides
10.four pairs of congruent angles
At an archeological site, the remains of two ancient step pyramids are congruent.
If ABCD @ EFGH, find each of the following. (Diagrams arehsm11gmse_0401_t02429
not to scale.)
11.AD
13.m∠GHE
15.EF
17.m∠DCB
12.GH
14.m∠BAD
16.BC
18.m∠EFG
B
45 ft 128
A
150
Lesson 4-1 Congruent Figures
F
C
45 ft
52
D
HSM11GMSE_0401_a02282
2nd pass 11-19-08
Durke
E
52
280 ft
G
128
335 ft
H
HSM11GMSE_0401_a02283
2nd pass 11-19-08
Durke
Explain Mathematical Ideas (1)(G) For Exercises 19 and 20, can you conclude that the
triangles are congruent? Justify your answers.
19.△TRK and △TUK 20.△SPQ and △TUV
T
7
S
R
U
V
5
Q
4
P
8
T
6
7
U
K
21.Given: AB } DC, ∠B ≅ ∠D,
Proof
AB ≅ DC, BC ≅ AD
Prove: △ABC ≅ △CDA
hsm11gmse_0401_t02432
B
C
hsm11gmse_0401_t02433
A
D
22.Evaluate Reasonableness (1)(B) Randall says he can use
the information in the figure to prove △BCD ≅ △DAB.
Is he correct? Explain.
C
D
hsm11gmse_0401_t02434 B
Connect Mathematical Ideas (1)(F) △ABC @ △DEF.
Find the measures of the given angles or the lengths of
the given sides.
A
23.m∠A = x + 10, m∠D = 2x
24.m∠B = 3y, m∠E = 6y - 12
25.BC = 3z + 2, EF = z + 6
26. AC = 7a + 5, DF = 5a hsm11gmse_0401_t02436
+9
27.If △DEF ≅ △LMN, which of the following must be a correct
congruence statement?
A.DE ≅ LN C.∠N ≅ ∠F
B.FE ≅ NL
D.∠M ≅ ∠F
E
L
M
D
F
N
Connect Mathematical Ideas (1)(F) Find the values of the variables.
28.
C
29.
M
D
3x
A
45
4 in.
B L 2t in.
ABC KLM
A
6x
30
C
hsm11gmse_0401_t02435
K
30.Complete in two different ways:
△JLM
≅ ? .
hsm11gmse_0401_t02438
B
ACD ACB
Z
M
J N
hsm11gmse_0401_t02439
L
R
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31.Given: AB # AD, BC # CD, AB ≅ CD, AD ≅ CB, AB } CD
A
B
D
C
Proof
Prove: △ABD ≅ △CDB
32.Analyze Mathematical Relationships (1)(F) Write a
congruence statement for two triangles. List the congruent
sides and angles.
33.Given: PR } TQ, PR ≅ TQ, PS ≅ QS, PQ bisects RT
P
hsm11gmse_0401_t02442
Proof
Prove: △PRS ≅ △QTS
34.Apply Mathematics (1)(A) The 225 cards in Tracy’s sports card
collection are rectangles of three different sizes. How could Tracy
quickly sort the cards?
T
S
35.Connect Mathematical Ideas (1)(F) The vertices of △GHJ
are G(-2, -1), H(-2, 3), and J(1, 3), and △KLM ≅ △GHJ.
If L and M have coordinates L(3, -3) and M(6, -3), how many
pairs of coordinates are possible for K? Find one such pair.
R
Q
36.a. How many quadrilaterals (convex and concave) with different shapes
or sizes
hsm11gmse_0401_t02441
can you make on a three-by-three geoboard? Sketch them. One is shown at
the right.
b. How many quadrilaterals of each type are there?
37.Given: ∠A ≅ ∠D, ∠B ≅ ∠E
Proof
Prove: ∠C ≅ ∠F
B
TEXAS Test Practice
D
A
C
E
F
hsm11gmse_0401_t05007
38.△HLN ≅ △GST , m∠H = 66, and m∠S = 42. What is m∠T ?
39.The measure of one angle in a triangle is 80. The other two angles are congruent.
What is the measure of each?
40.What is the number of feet in the perimeter of a square with side length 7 ft?
152
Lesson 4-1 Congruent Figures
hsm11gmse_0401_t02443
4-2 Triangle Congruence by SSS and SAS
TEKS FOCUS
VOCABULARY
•Number sense – the understanding
TEKS (6)(B) Prove two triangles are congruent by applying the Side-AngleSide, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg
congruence conditions.
of what numbers mean and how they
are related
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental
math, estimation, and number sense as appropriate, to solve problems.
Additional TEKS (1)(E), (1)(F), (1)(G), (5)(A), (5)(C)
ESSENTIAL UNDERSTANDING
You can prove that two triangles are congruent without having to show that all
corresponding sides and angles are congruent. In this lesson, you will prove triangles
congruent by using (1) three pairs of corresponding sides and (2) two pairs of
corresponding sides and one pair of corresponding angles.
Postulate 4-1 Side-Side-Side (SSS) Postulate
Postulate
If the three sides of one triangle
are congruent to the three sides
of another triangle, then the two
triangles are congruent.
If . . .
AB ≅ DE, BC ≅ EF , AC ≅ DF
B
E
A
D
C
Then . . .
△ABC ≅ △DEF
F
Postulate 4-2 Side-Angle-Side (SAS) Postulate
Postulate
If two sides and the included
angle of one triangle are
congruent to two sides and
the included angle of another
triangle, then the two triangles
are congruent.
If . . .
hsm11gmse_0402_t05017
AB ≅ DE, ∠A ≅ ∠D, AC ≅ DF
B
Then . . .
△ABC ≅ △DEF
E
A
D
C
F
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153
Problem 1
TEKS Process Standard (1)(C)
Building Triangles
A Select a tool to build a triangle with sides of lengths 3 in., 5 in., and 6 in.
Compare your triangle with your classmates’ triangles.
What are
corresponding sides?
Corresponding sides are
the sides that match and
are in the same position.
Corresponding sides of
congruent triangles have
the same length.
To build a triangle with the given lengths, you might
select a real object, such as a straw. Cut the straw into
pieces of the correct lengths. Thread some string
through the three pieces of straw, in any order, as
shown. Bring the ends of the string together and tie
them to hold your triangle in place.
B Make a conjecture about two triangles in which three sides
of one triangle are congruent to three sides of the other triangle.
Both triangles have the same size and shape. Each triangle fits exactly on
top of the other triangle.
Conjecture: Triangles with congruent corresponding sides are congruent.
Problem 2
TEKS Process Standard (1)(E)
Proof Using SSS
You have two pairs
of congruent sides.
What else do you
need?
You need a third pair of
congruent corresponding
sides. Notice that
the triangles share a
common side, LN.
N
L
M
Given: LM ≅ NP, LP ≅ NM
Prove: △LMN ≅ △NPL
LM ≅ NP
LN ≅ LN
LP ≅ NM
Given
Reflexive Prop. of ≅
Given
△LMN ≅ △NPL
SSS
154
P
Lesson 4-2 Triangle Congruence by SSS and SAS
hsm11gmls_0402_t05019
Problem 3
Using SAS
Do you need another
pair of congruent
sides?
Look at the diagram.
The triangles share DF.
So you already have two
pairs of congruent sides.
E
F
What other information do you need to prove
△DEF @ △FGD by SAS? Explain.
The diagram shows that EF ≅ GD. Also, DF ≅ DF by
the Reflexive Property of Congruence. To prove that
△DEF ≅ △FGD by SAS, you must have congruent
included angles. You need to know that ∠EFD ≅ ∠GDF.
D
G
Problem 4
Identifying Congruent Triangles
NLINE
HO
ME
RK
O
What should you
look for first, sides or
angles?
Start with sides. If you
have three pairs of
congruent sides, use SSS.
If you have two pairs of
congruent sides, look
for a pair of congruent
included angles.
WO
hsm11gmse_0402_t05022
Would you use SSS or SAS to prove the triangles congruent? If there is not enough
information to prove the triangles congruent by SSS or SAS, write not enough
information. Explain your answer.
A B Use SAS because two pairs of
corresponding sides and their included
angles are congruent.
C hsm11gmse_0402_t05025
There is not enough information; two pairs
of corresponding sides are congruent, but
one of the angles is not the included angle.
D hsm11gmse_0402_t05026
Use SSS because three pairs of
corresponding sides are congruent.
Use SSS or SAS because all three pairs of
corresponding sides and a pair of included
angles (the vertical angles) are congruent.
hsm11gmse_0402_t05027
hsm11gmse_0402_t05028
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1.
Use Representations to Communicate Mathematical
Ideas (1)(E) Copy the flow chart and complete the proof.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Given: JK ≅ LM, JM ≅ LK
JK ≅ LM
K
JM ≅ LK
Given
a.
c.
L
J
Prove: △JKM ≅ △LMK
M
KM ≅ KM
b.
≅ d.
hsm11gmse_0402_t02639.ai
SSS
PearsonTEXAS.com
hsm11gmse_0402_t02640.ai
155
2.
Select Tools to Solve Problems (1)(C) Consider the following conjecture.
If two triangles have the same perimeter, then the triangles are congruent.
a.Select a real object that you can use to test the conjecture. Explain your choice.
b.Is the conjecture true? If not, make a new conjecture based on your results.
Explain your reasoning.
3.
Explain Mathematical Ideas (1)(G) At least how many triangle measurements
must you know in order to guarantee that all triangles built with those
measurements will be congruent? Explain your reasoning.
4.
Given: IE ≅ GH, EF ≅ HF,
5.Given: WZ ≅ ZS ≅ SD ≅ DW
Proof
Proof
Prove: △WZD ≅ △SDZ
F is the midpoint of GI Prove: △EFI ≅ △HFG
W
G
E
Z
F
D
H
I
S
What other information, if any, do you need to prove the two triangles
congruent by SAS? Explain.
6.
hsm11gmse_0402_t02641.ai
G
N
L
T
7. hsm11gmse_0402_t02642.ai
U
T
W
R
M
Q
V
S
8.
Evaluate Reasonableness (1)(B) You and a friend are cutting triangles out of
felt for an art project. You want all the triangles to be congruent. Your friend tells
you that each triangle should have two 5-in.hsm11gmse_0402_t02644.ai
sides and a 40° angle. If you follow
hsm11gmse_0402_t02643.ai
this rule, will all your felt triangles be congruent? Explain.
Can you prove the triangles congruent? If so, write the congruence statement and
name the postulate you would use. If not, write not enough information and tell
what other information you would need.
9.
A
10.
G
N
R
T
Y
H
W
K
P
D
11.
J
E
S
F
V
T
hsm11gmse_0402_t02651.ai
hsm11gmse_0402_t02650.ai
hsm11gmse_0402_t02649.ai
156
Lesson 4-2 Triangle Congruence by SSS and SAS
12.Use Representations to Communicate Mathematical
Ideas (1)(E) Sierpinski’s triangle is a famous geometric pattern.
To draw Sierpinski’s triangle, start with a single triangle and connect
the midpoints of the sides to draw a smaller triangle. If you repeat
this pattern over and over, you will form a figure like the one shown.
This particular figure started with an isosceles triangle. Are the
triangles outlined in red congruent? Explain.
13.Create Representations to Communicate
Mathematical Ideas (1)(E) Use a straightedge to draw
any triangle JKL. Then construct △MNP ≅ △JKL using
the given postulate.
a.SSS
b.SAS
14.Analyze Mathematical Relationships (1)(F) Suppose GH ≅ JK , HI ≅ KL, and
∠I ≅ ∠L. Is △GHI congruent to △JKL? Explain.
15.Given: FG } KL, FG ≅ KL
Proof
Prove: △FGK ≅ △KLF
F
G
16.Given: AB # CM, AB # DB, CM ≅ DB,
Proof
M is the midpoint of AB.
Prove: △AMC ≅ △MBD
D
L
B
C
K
M
A
hsm11gmse_0402_t02654.ai
TEXAS Test Practice
hsm11gmse_0402_t02655.ai
Y
17.What additional information do you need to prove that
△VWY ≅ △VWZ by SAS?
A.YW ≅ ZW C.∠Y ≅ ∠Z
B.∠WVY ≅ ∠WVZ
D.VZ ≅ VY
V
18.The measures of two angles of a triangle are 43 and 38. What is
the measure of the third angle?
F.
9
G.81
H.99
W
Z
J.100
hsm11gmse_0402_t02658.ai
19.Which method would you use to find the inverse of a conditional statement?
A.Negate the hypothesis only.
C.Negate the conclusion only.
B.Switch the hypothesis and D.Negate both the hypothesis and
the conclusion. the conclusion.
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157
4-3 Triangle Congruence by ASA and AAS
TEKS FOCUS
VOCABULARY
•Analyze – closely examine objects, ideas, or relationships to
TEKS (6)(B) Prove two triangles are congruent by
applying the Side-Angle-Side, Angle-Side-Angle, SideSide-Side, Angle-Angle-Side, and Hypotenuse-Leg
congruence conditions.
learn more about their nature.
TEKS (1)(F) Analyze mathematical relationships to
connect and communicate mathematical ideas.
Additional TEKS (1)(A), (1)(D), (1)(G)
ESSENTIAL UNDERSTANDING
You can prove that two triangles are congruent without having to show that all
corresponding parts are congruent. In this lesson, you will prove triangles congruent
by using one pair of corresponding sides and two pairs of corresponding angles.
Postulate 4-3 Angle-Side-Angle (ASA) Postulate
Postulate
If two angles and the included
side of one triangle are
congruent to two angles and
the included side of another
triangle, then the two triangles
are congruent.
If . . .
∠A ≅ ∠D, AC ≅ DF , ∠C ≅ ∠F
B
E
A
C
D
Then . . .
△ABC ≅ △DEF
F
Theorem 4-2 Angle-Angle-Side
(AAS) Theorem
hsm11gmse_0403_t05033.ai
Theorem
If two angles and a nonincluded
side of one triangle are
congruent to two angles and
the corresponding nonincluded
side of another triangle, then the
triangles are congruent.
If . . .
∠A ≅ ∠D, ∠B ≅ ∠E, AC ≅ DF
B
F
C
A
D
Then . . .
△ABC ≅ △DEF
E
You will prove Theorem 4-2 in Exercise 15.
hsm11gmse_0403_t05038.ai
158
Lesson 4-3 Triangle Congruence by ASA and AAS
Problem 1
TEKS Process Standard (1)(F)
Using ASA
O
Which two triangles are congruent by ASA? Explain.
W
U
From the diagram you know
• ∠U ≅ ∠E ≅ ∠T
• ∠V ≅ ∠O ≅ ∠W
• UV ≅ EO ≅ AW
To use ASA, you need two pairs
of congruent angles and a pair of
included congruent sides.
N
E
S
V
T
A
You already have pairs of congruent angles. So,
identify the included side for each triangle and see
whether it has a congruence marking.
hsm11gmse_0403_t05036.ai
In △SUV, UV is included between ∠U and ∠V and has a congruence
marking. In △NEO, EO is included between ∠E and ∠O and has a
congruence marking. In △ATW, TW is included between ∠T and ∠W but
does not have a congruence marking.
Since ∠U ≅ ∠E, UV ≅ EO, and ∠V ≅ ∠O, △SUV ≅ △NEO.
Problem 2
TEKS Process Standard (1)(A)
Proof Writing a Proof Using ASA
Recreation Members of a teen organization are
building a miniature golf course at your town’s youth
center. The design plan calls for the first hole to have
two congruent triangular bumpers. Prove that the
bumpers on the first hole, shown at the right, meet
the conditions of the plan.
Can you use a plan
similar to the plan in
Problem 1?
Yes. Use the diagram to
identify the included side
for the marked angles in
each triangle.
A
Given: AB ≅ DE, ∠A ≅ ∠D, ∠B and ∠E are
right angles
B
Prove: △ABC ≅ △DEF
Proof: ∠B ≅ ∠E because all right angles are
congruent, and you are given that ∠A ≅ ∠D.
AB and DE are included sides between the two
pairs of congruent angles. You are given that
AB ≅ DE. Thus, △ABC ≅ △DEF by ASA.
D
E
C
F
PearsonTEXAS.com
159
Problem 3
Proof Writing a Proof Using AAS
How does information
about parallel sides
help?
You will need another
pair of congruent angles
to use AAS. Think back
to what you learned
in Topic 3. WR is a
transversal here.
M
R
Given: ∠M ≅ ∠K, WM } RK
Prove: △WMR ≅ △RKW
W
Statements
K
Reasons
1) ∠M ≅ ∠K
1) Given
2) WM } RK
2) Given
3) ∠MWR ≅ ∠KRW
hsm11gmse_0403_t05041.ai
3) If lines are } , then alternate interior ⦞ are ≅.
4) WR ≅ WR
4) Reflexive Property of Congruence
5) △WMR ≅ △RKW
5) AAS
Problem 4
Determining Whether Triangles Are Congruent
B
Multiple Choice Use the diagram at the right. Which of the
following statements best represents the answer and justification
to the question, “Is △BIF @ △UTO?”
Yes, the triangles are congruent by ASA.
No, FB and OT are not corresponding sides.
Yes, the triangles are congruent by AAS.
Can you eliminate
any of the choices?
Yes. If △BIF @ △UTO
then ∠B and ∠U would
be corresponding angles.
You can eliminate
choice D.
I
F
No, ∠B and ∠U are not corresponding angles.
U
The diagram shows that two pairs of angles and one pair of sides are
congruent. The third pair of angles is congruent by the Third Angles
Theorem. To prove these triangles congruent, you need to satisfy
ASA or AAS.
ASA and AAS both fail because FB and TO are not included
between the same pair of congruent corresponding angles, so they
are not corresponding sides. The triangles are not necessarily
congruent. The correct answer is B.
O
T
hsm11gmse_0403_t05044.ai
160
Lesson 4-3 Triangle Congruence by ASA and AAS
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Determine whether the triangles must be congruent. If so, name the postulate or
theorem that justifies your answer. If not, explain.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
T
2.
M
3.
W
V
U
P
R
N
O
S
4.Given: ∠FJG ≅ ∠HGJ, FG } JH
Proof
hsm11gmse_0403_t02698.ai
△FGJ ≅ △HJG
Prove: F
Z
Y
5.Given: PQ # QS, RS # SQ,
Proof
T is the midpoint of PR
hsm11gmse_0403_t02699.ai
hsm11gmse_0403_t02700
Prove: △PQT ≅ △RST
G
R
Q
J
T
H
S
P
6.
Evaluate Reasonableness (1)(B) While helping your
family clean out the attic, you find the piece of
hsm11gmse_0403_t02702.ai
paper
shown at the right. The paper contains clues
hsm11gmse_0403_t02697.ai
to locate a time capsule buried in your backyard.
The maple tree is due east of the oak tree in your
backyard. Will the clues always lead you to the
correct spot? Explain.
7.
Connect Mathematical Ideas (1)(F) Anita says
that you can rewrite any proof that uses the AAS
Theorem as a proof that uses the ASA Postulate.
Do you agree with Anita? Explain.
8.
Justify Mathematical
Arguments (1)(G) Can you prove
that the triangles at the right are
congruent? Justify your answer.
9.Given: ∠N ≅ ∠P, MO ≅ QO
Proof
Prove: △MON ≅ △QOP
Proof
M
hsm11gmse_0403_t02703.ai
Prove: △BDH ≅ △FDH
N
10.Given: ∠1 ≅ ∠2, and
HSM11GMSE_0403_a02291
DH bisects ∠BDF
3rd pass 12-22-08
O
Q
1
P
B
hsm11gmse_0403_t02701.ai
Durke
D
H
2
F
PearsonTEXAS.com
hsm11gmse_0403_t02706
161
11.Given: AB } DC, AD } BC
A
B
Proof
Prove: △ABC ≅ △CDA
C
D
12.Create Representations to Communicate Mathematical Ideas (1)(E) Draw two
noncongruent triangles that have two pairs of congruent angles and one pair of
congruent sides.
hsm11gmse_0403_t02708.ai
13.Given AD } BC and AB } DC, name
as many pairs of congruent
triangles as you can.
B
C
E
14.Create Representations to Communicate Mathematical Ideas (1)(E) Use a straightedge to draw a triangle. Label it △JKL. Construct
△MNP ≅ △JKL so that the triangles are congruent by ASA.
A
D
15.Prove the Angle-Angle-Side Theorem (Theorem 4-2). Use the diagram next to it
on page 158.
16.In △RST at the right, RS = 5, RT = 9, and m∠T = 30. Show that
there is no SSA congruence rule by constructing △UVW with
UV = RS, UW = RT , and m∠W = m∠T , but with △UVW R △RST .
R
hsm11gmse_0403_t02709.ai
9
5
30
S
TEXAS Test Practice
hsm11gmse_0403_t02710.ai
17.Suppose RT ≅ ND and ∠R ≅ ∠N. What additional information do you need to
prove that △RTJ ≅ △NDF by ASA?
A.∠T ≅ ∠D
C.∠J ≅ ∠D
B.∠J ≅ ∠F D.∠T ≅ ∠F
18.You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. Each photo
is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest
number of photos you can fit on your poster?
F.
4
H.32
G.24
J.36
19.Write the converse of the true conditional statement below. Then determine
whether the converse is true or false.
If you are less than 18 years old, then you are too young to vote in the United
States.
162
Lesson 4-3 Triangle Congruence by ASA and AAS
T
Technology Lab
Use With Lesson 4-3
Exploring AAA and SSA
teks (5)(A), (1)(F)
So far, you know four ways to conclude that two triangles are congruent—SSS, SAS,
ASA, and AAS. It is good mathematics to wonder about the other two possibilities.
1
>
>
Construct Use geometry software to construct AB and AC . Construct BC to
>
>
form △ABC. Construct a line parallel to BC that intersects AB and AC at
points D and E to form △ADE.
B
Investigate Are the three angles of △ABC congruent to the three angles of
△ADE? Manipulate the figure to change the positions of DE and BC. Do the
corresponding angles of the triangles remain congruent? Are the two triangles
congruent? Can the two triangles be congruent?
2
>
D
E
A
C
>
Construct Construct AB . Draw a circle with center C that intersects AB at two
hsm11gmse_04fa_t05104.ai
points. Construct AC. Construct point E on the circle and construct CE.
>
Investigate Move point E around the circle until E is on AB and forms
>
△ACE. Then move E on the circle to the other point on AB to form
another △ACE.
D
C
Compare AC, CE, and m∠A in the two triangles. Are two sides and a
nonincluded angle of one triangle congruent to two sides and a nonincluded
angle of the other triangle? Are the triangles congruent? If you change the
measure of ∠A and the size of the circle, do you get the same results?
A
E
B
hsm11gmse_04fa_t05105.ai
Exercises
1.Make a Conjecture Based on your first investigation above, can you prove
triangles congruent using AAA? Explain.
For Exercises 2–4, use what you learned in your second investigation above.
2.Make a Conjecture Can you prove triangles congruent using SSA? Explain.
>
3.Manipulate the figure so that ∠A is obtuse. Can the circle intersect AB twice
to form two triangles? Would SSA work if the congruent angles were obtuse?
Explain.
4.Suppose you are given CE, AC, and ∠A. What must be true about CE, AC, and
m∠A so that you can construct exactly one △ACE? (Hint: Consider cases.)
PearsonTEXAS.com
163
4-4 Using Corresponding Parts of
Congruent Triangles
TEKS FOCUS
VOCABULARY
•Justify – explain with logical reasoning. You can justify a
TEKS (6)(B) Prove two triangles are congruent by
applying the Side-Angle-Side, Angle-Side-Angle,
Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions.
mathematical argument.
•Argument – a set of statements put forth to show the truth or
falsehood of a mathematical claim
TEKS (1)(G) Display, explain, and justify
mathematical ideas and arguments using precise
mathematical language in written or oral
communication.
ESSENTIAL UNDERSTANDING
If you know two triangles are congruent, then you know that every pair of their
corresponding sides and angles is also congruent.
Problem 1
Proof Proving Parts of Triangles Congruent
Given: ∠KBC ≅ ∠ACB, ∠K ≅ ∠A
Prove: KB ≅ AC
K
In the diagram, which
congruent pair is not
marked?
The third angles of both
triangles are congruent.
But there is no AAA
congruence rule. So, find
a congruent pair of sides.
164
B
KBC ACB
Given
C
A
BC BC
Reflexive Property of hsm11gmse_0404_t02447
K A
Given
KBC ACB
AAS Theorem
Lesson 4-4 Using Corresponding Parts of Congruent Triangles
KB AC
Corresp. parts of are .
Problem 2
TEKS Process Standard (1)(G)
Proof Proving Triangle Parts Congruent to Measure Distance
Which congruency
rule can you use?
You have information
about two pairs of
angles. Guess-andcheck AAS and ASA.
STEM
Measurement Thales, a Greek philosopher, is said to have developed a method
to measure the distance to a ship at sea. He made a compass by nailing two sticks
together. Standing on top of a tower, he would hold one stick vertical and tilt
the other until he could see the ship S along the line of the tilted stick. With this
compass setting, he would find a landmark L on the shore along the line of the
tilted stick. How far would the ship be from the base of the tower?
Given: ∠TRS and ∠TRL are right angles, ∠RTS ≅ ∠RTL
Prove: RS ≅ RL
T
S
L
R
Statements
Reasons
1) ∠RTS ≅ ∠RTL
1) Given
2) TR ≅ TR
2) Reflexive Property of Congruence
3) ∠TRS and ∠TRL are right angles.
3) Given
4) ∠TRS ≅ ∠TRL
4) All right angles are congruent.
5) △TRS ≅ △TRL
5) ASA Postulate
6) RS ≅ RL
s are ≅.
6) Corresponding parts of ≅ △
The distance between the ship and the base of the tower would be the same as the
distance between the base of the tower and the landmark.
PearsonTEXAS.com
165
HO
ME
RK
O
NLINE
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PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Scan page for a Virtual Nerd™ tutorial video.
1.
Explain Mathematical Ideas (1)(G) Tell why the two triangles are
congruent. Give the congruence statement. Then list all the
other corresponding parts of the triangles that are congruent.
L
K
2.Given: ∠ABD ≅ ∠CBD,
Proof
∠BDA ≅ ∠BDC
3. Given: OM ≅ ER, ME ≅ RO
O
N
Prove: ∠M ≅ ∠R
O
hsm11gmse_0404_t02457
B
C
A
J
Proof
Prove: AB ≅ CB
M
R
M
E
D
4.
Justify Mathematical Arguments (1)(G) A balalaika is
a stringed instrument. Prove that the bases of the
hsm11gmse_0404_t02463
balalaikas
are congruent.
hsm11gmse_0404_t02461
Given: RA ≅ NY , ∠KRA ≅ ∠JNY , ∠KAR ≅ ∠JYN
R
Prove: KA ≅ JY
Proof: It is given that two angles and the included
side of one triangle are congruent to two angles
and the included side of the other. So,
a. ? ≅ △JNY by b. ? . KA ≅ JY because c. ? .
5.Given: ∠SPT ≅ ∠OPT , Proof SP ≅ OP
T
K
A
J
Y
6.Given: YT ≅ YP, ∠C ≅ ∠R,
∠T ≅ ∠P
Proof
Prove: ∠S ≅ ∠O
S
N
Prove: CT ≅ RP
O
R
C
Y
P
P
T
Analyze Mathematical Relationships (1)(F) Copy and mark
the figure to show the given information. Explain how you
wouldhsm11gmse_0404_t02464
prove jP @ jQ.
K
hsm11gmse_0404_t02465
7.
Given: PK ≅ QK , KL bisects ∠PKQ
8.
Given: KL is the perpendicular bisector of PQ.
9.
Given: KL # PQ, KL bisects ∠PKQ
166
Lesson 4-4 Using Corresponding Parts of Congruent Triangles
P
L
Q
C
10.Justify Mathematical Arguments (1)(G) The construction of a line
perpendicular to line / through point
< P> on line / is shown.
Explain why you can conclude that CP is perpendicular to /.
P
A
11.The construction of ∠B congruent to given ∠A is shown.
AD ≅ BF because they are congruent radii. DC ≅ FE
because both arcs have the same compass settings.
Explain why you can conclude that ∠A ≅ ∠B.
E
C
hsm11gmse_0404_t02468
D
A
Prove: AB ≅ CD
K
C
E
A
F
D
F
B
13.Given: JK } QP, JK ≅ PQ
12.Given: BE # AC, DF # AC,
Proof
BE ≅ DF , AF ≅ CE
B
B
Proof
Prove: KQ bisects JP
hsm11gmse_0404_t02474
P
M
J
Q
14.Apply Mathematics (1)(A) Rangoli is a colorful design pattern
drawn outside houses in India, especially during festivals.
Vina plans to use the pattern at the right as the base of her design.
In this pattern, RU , SV , and QT bisect eachhsm11gmse_0404_t02479
other at O.
hsm11gmse_0404_t02476
RS = 6, RU = 12, RU ≅ SV , ST } RU , and RS } QT .
What is the perimeter of the hexagon?
In the diagram at the right, BA @ KA and BE @ KE.
15.Prove: S is the midpoint of BK .
Proof
16. Prove: BK # AE
A
K
S
E
Proof
B
TEXAS Test Practice
hsm11gmse_0404_t02481
For Exercises 17 and 18, use the diagram at the right. TM # BD
and TM bisects jBTD and j ATC.
B
A
M
C
D
17.Suppose BD = 17 and AM = 5. What is the length of CD?
18.Suppose m∠ATC = 64, and m∠BTA = 16. What is m∠B?
19.Two parallel lines q and s are cut by a transversal t. ∠1 and ∠2 are a pair of alternate
interior angles and m∠2 = 38. ∠1 and ∠3 are vertical angles. What is m∠3?
T
20.△ABC has vertices A(1, 9), B(4, 3), and C(x, 6). For what value of x is △ABC a right
triangle with right ∠B?
hsm11gmse_0404_t02482
PearsonTEXAS.com
167
4-5 Isosceles and Equilateral Triangles
TEKS FOCUS
VOCABULARY
TEKS (5)(C) Use the constructions of
congruent segments, congruent angles,
angle bisectors, and perpendicular
bisectors to make conjectures about
geometric relationships.
TEKS (1)(C) Select tools, including
real objects, manipulatives, paper and
pencil, and technology as appropriate,
and techniques, including mental
math, estimation, and number sense as
appropriate, to solve problems.
Additional TEKS (5)(A), (6)(B), (6)(D)
•Base of an isosceles triangle – the non-congruent side of an isosceles
triangle
•Base angles of an isosceles triangle – the two angles in an isosceles triangle
that are formed by the intersection of a leg and the base
•Corollary – a theorem that can be proved easily using another theorem
•Legs of an isosceles triangle – the congruent sides of an isosceles triangle
•Vertex angle of an isosceles triangle – the angle in an isosceles triangle
formed by the two congruent legs
•Number sense – the understanding of what numbers mean and how they
are related
ESSENTIAL UNDERSTANDING
The sides and angles of isosceles and equilateral triangles have special relationships.
Theorem 4-3 Isosceles Triangle Theorem
Theorem
If two sides of a triangle are
congruent, then the angles
opposite those sides are
congruent.
If . . .
AC ≅ BC
Then . . .
∠A ≅ ∠B
C
A
C
B
A
B
For a proof of Theorem 4-3, see Problem 2.
hsm11gmse_0405_t02727
Theorem 4-4 Converse ofhsm11gmse_0405_t02726
the Isosceles Triangle Theorem
Theorem
If two angles of a triangle are
congruent, then the sides
opposite those angles are
congruent.
If . . .
∠A ≅ ∠B
Then . . .
AC ≅ BC
C
A
C
A
B
B
You will prove Theorem 4-4 in Exercise 16.
hsm11gmse_0405_t02729
168
Lesson 4-5 Isosceles and Equilateral Triangles
hsm11gmse_0405_t02730
Theorem 4-5
Theorem
If a line bisects the vertex angle
of an isosceles triangle, then the
line is also the perpendicular
bisector of the base.
If . . .
AC ≅ BC and
∠ACD ≅ ∠BCD
Then . . .
CD # AB and
AD ≅ BD
C
A
D
B
C
A
D
B
You will prove Theorem 4-5 in Exercise 10.
Corollary to Theorem 4-3 hsm11gmse_0405_t02734.ai hsm11gmse_0405_t02
Corollary
If a triangle is equilateral, then
the triangle is equiangular.
Y
If . . .
XY ≅ YZ ≅ ZX
Then . . .
∠X ≅ ∠Y ≅ ∠Z
Z
X
Y
Z
X
Corollary to Theorem 4-4
Corollary
If a triangle is equiangular,
then the triangle is equilateral.
If . . .
∠X ≅ ∠Y ≅ ∠Z
Then . . .
Y
Z
X
Problem 1
Y
hsm11gmse_0405_t02736.ai
XY ≅ YZ ≅ ZX hsm11gmse_0405_t027
X
Z
TEKS Process Standard (1)(C)
hsm11gmse_0405_t02737.ai hsm11gmse_0405_t027
Using Constructions of Congruent Segments
Construct congruent segments to make a conjecture about the angles opposite the
A congruent sides in an isosceles triangle.
Step 1 Construct an isosceles ∆ABC on tracing paper, with AC ≅ BC.
C
A
B
A
B
A
Step 2
Fold the paper so that the two congruent sides fit exactly
one on top of the other. Crease the paper. Notice that
hsm11gmse_04fa_t02587.ai
∠A and ∠B appear to be congruent.
Conjecture: Angles opposite the congruent sides in an
isosceles triangle are congruent.
B
B
How can folding a
piece of paper help
you tell if two angles
are congruent?
When folding the paper,
congruent angles will fit
exactly one on top of the
other.
C
A
continued on next page ▶
PearsonTEXAS.com 169
hsm11gmse_04fa_t02588.ai
Problem 1
continued
Construct congruent angles to make a conjecture about the sides opposite
B congruent angles in a triangle.
Step 1Draw ∠ABC on tracing paper. Then construct ∠CBD congruent to
∠ABC so that CD intersects AB, resulting in a triangle.
A
B
D
C
A
B
C
Step 2Fold the paper so that the two congruent angles fit exactly on top of each
other. Notice that the sides of the triangle opposite the congruent angles
appear to be congruent.
Conjecture: Sides opposite the congruent angles in a triangle are congruent.
Problem 2
Proof
How are the sides and
angles of an isosceles
triangle related?
The congruent sides of
an isosceles triangle are
its legs, and the third
side is its base. The two
congruent legs form the
vertex angle, while the
other two angles are the
base angles.
Proving the Isosceles Triangle Theorem
X
Begin with isosceles △XYZ with XY ≅ XZ. Draw XB, the bisector
of vertex angle jYXZ.
1 2
Given: XY ≅ XZ, XB bisects ∠YXZ
Y
Prove: ∠Y ≅ ∠Z
Statements
B
Z
Reasons
1) XY ≅ XZ
1) Given
2) ∠1 ≅ ∠2
2) Definition of angle bisector
3) XB ≅ XB
3) Reflexive Property of Congruence
4) △XYB ≅ △XZB
4) SAS Postulate
5) ∠Y ≅ ∠Z
s are ≅.
5) Corresponding parts of ≅ △
hsm11gmse_0405_t02728
Problem 3
What are you looking
for in the diagram?
To use the Isosceles
Triangle theorems, you
need a pair of congruent
angles or a pair of
congruent sides.
170
Using the Isosceles Triangle Theorem and its Converse
A Is AB congruent to CB? Explain.
B
Yes. Since ∠C ≅ ∠A, AB ≅ CB by the Converse of the
Isosceles Triangle Theorem.
D
B Is jA congruent to jDEA? Explain.
Yes. Since AD ≅ ED, ∠A ≅ ∠DEA by the Isosceles Triangle
Theorem.
Lesson 4-5 Isosceles and Equilateral Triangles
A
E
C
hsm11gmse_0405_t02731.ai
Problem 4
Using Algebra
What does the
diagram tell you?
Since AB ≅ CB, △ABC
is isosceles. Since
∠ABD ≅ ∠CBD, BD
bisects the vertex angle
of the isosceles triangle.
A
What is the value of x?
x
D
Since BD bisects ∠ABC, you know by Theorem 4-5
that BD # AC. So m∠BDC = 90.
m∠C + m∠BDC + m∠DBC = 180
54 + 90 + x = 180
x = 36 C
3 6
Triangle Angle-Sum Theorem
Substitute.
Subtract 144 from each side.
Problem 5
B
54
Since AB ≅ CB, by the Isosceles Triangle Theorem,
∠A ≅ ∠C. So m∠C = 54.
. . . . . . .
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
hsm11gmse_0405_t02735.ai
TEKS Process Standard (1)(G)
Finding Angle Measures
Design What are the measures of jA, jB, and jADC in the
photo image at the right?
D
The triangles are equilateral,
so they are also equiangular.
Find the measure of each
angle of an equilateral triangle.
Let a = measure of one angle.
3a = 180
a = 60
∠A and ∠B are both angles
in an equilateral triangle.
m∠A = m∠B = 60
Use the Angle Addition
Postulate to find the
measure of ∠ADC.
m∠ADC = m∠ADE + m∠CDE
Both ∠ADE and ∠CDE are
angles in an equilateral
triangle. So m∠ADE = 60
and m∠CDE = 60. Substitute
into the above equation
and simplify.
A
C
E
B
m∠ADC = 60 + 60
m∠ADC = 120
PearsonTEXAS.com
171
HO
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NLINE
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Complete each statement. Explain why it is true.
V
1.
VT ≅ ?
For additional support when
completing your homework,
go to PearsonTEXAS.com.
2.
UT ≅ ? ≅ YX
3.
VU ≅ ?
U
4.
∠VYU ≅ ?
T
Y
X
W
A
5.
Justify Mathematical Arguments (1)(G) A builder using
the truss shown at the right claims that ∠ACB will have
the same measure as ∠ADB. AC and AD represent
hsm11gmse_0405_t02743.ai
identical beams, and AB bisects ∠CAD. Is the
builder
correct? Justify your answer.
C
B
king
post
truss
Connect Mathematical Ideas (1)(F) Find the values of x and y.
6.
7.
x
100
50 y 8.
52
x
4
x
hsm11gmse_0401_t02427
y
110
y
Mathematics (1)(A) Each face of the Great Pyramid at Giza is an isosceles
triangle with a 76° vertex angle. What are the measures of the base angles?
STEM9.Apply
hsm11gmse_0405_t02746.ai
hsm11gmse_0405_t02745.ai
10.Prove
Theorem 4-5. Use the diagram
next to it on page 169.
hsm11gmse_0405_t02744.ai
Given isosceles △JKL with base JL, find each value.
K
11.If m∠L = 58, then m∠LKJ = ? .
12.If JL = 5, then ML = ? .
13.If m∠JKM = 48, then m∠J = ? .
14.If m∠J = 55, then m∠JKM = ? .
J
L
M
15.Analyze Mathematical Relationships (1)(F) A triangle has angle measures
x + 15, 3x - 35, and 4x. What type of triangle is it? Be as specific as possible.
Justify your answer.
hsm11gmse_0405_t02748.ai
16.Supply the missing information in this statement of the Converse of the Isosceles
Proof Triangle Theorem. Then write a proof.
R
Begin with △PRQ with ∠P ≅ ∠Q.
Draw a. ? , the bisector of ∠PRQ.
Given: ∠P ≅ ∠Q, b. ? bisects ∠PRQ
Prove: PR ≅ QR
172
P
S
Q
Lesson 4-5 Isosceles and Equilateral Triangles
hsm11gmse_0405_t02750.ai
E
G
D
F
attic fra
STEM
17. a.Apply Mathematics (1)(A) In the diagram
at the right, what type of triangle is formed
by the cables of the same height and the
ground?
b.What are the two different base lengths
of the triangles?
c.How is the tower related to each of
the triangles?
800 ft
600 ft
Cables
18.Analyze Mathematical Relationships (1)(F) The length of the base of an isosceles
triangle is x. The length of a leg is 2x - 5. The
perimeter of the triangle is 20. Find x.
For each pair of points, there are six points that
could be the third vertex of an isosceles right
triangle. Find the coordinates of each point.
19.(4, 0) and (0, 4)
Radio
tower
1009 ft tall
1000 ft
400 ft
200 ft
0 ft
450 ft
20.(0, 0) and (5, 5)
21.(2, 3) and (5, 6)
550 ft
HSM11GMSE_0405_a02297
C pass 12-22 -08
22.Create Representations to Communicate Mathematical Ideas (1)(E) 3rd
Durke
Use △ABC at the right.
A
a.Construct a right triangle with one leg congruent to AB and hypotenuse
congruent to BC .
b.Construct a right triangle with one leg congruent to AC and hypotenuse
congruent to BC .
c.Draw a new right triangle with different side lengths than △ABC. Repeat parts
(a) and (b) for your new right triangle.
d.Use your results to make a conjecture about congruence criteria for right
triangles.
B
TEXAS Test Practice
23.In isosceles △ABC, the vertex angle is ∠A. What can you prove?
A.AB = CB
B.m∠B = m∠C
C.∠A ≅ ∠B
D. BC ≅ AC
24.What is the exact area of the base of a circular swimming pool with diameter 16 ft?
F.
1018.29 ft2
G.1018.3 ft2
H.64p ft2
J.256p ft2
25.Suppose △ABC and △DEF are nonright triangles. If ∠B ≅ ∠E and AB ≅ DE,
what else do you need to know to prove △ABC ≅ △DEF ? Explain.
PearsonTEXAS.com
173
4-6 Congruence in Right Triangles
TEKS FOCUS
VOCABULARY
•Hypotenuse – the side opposite the right angle
•Legs of a right triangle – the two sides other than the hypotenuse
TEKS (6)(B) Prove two triangles are congruent by
applying the Side-Angle-Side, Angle-Side-Angle,
Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions.
in a right triangle
•Analyze – closely examine objects, ideas, or relationships to learn
TEKS (1)(F) Analyze mathematical relationships
to connect and communicate mathematical ideas.
more about their nature
ESSENTIAL UNDERSTANDING
You can prove that two triangles are congruent without having to show that all
corresponding sides and angles are congruent. In this lesson, you will prove right
triangles congruent by using one pair of right angles, a pair of hypotenuses, and a
pair of legs.
Theorem 4-6 Hypotenuse-Leg (HL) Theorem
Theorem
If the hypotenuse and a leg of
one right triangle are congruent
to the hypotenuse and a leg of
another right triangle, then the
triangles are congruent.
If . . .
s,
△PQR and △XYZ are right △
PR ≅ XZ, and PQ ≅ XY
X
P
Q
R
Then . . .
△PQR ≅ △XYZ
Z
Y
For a proof of Theorem 4-6, see the Reference section on page 683.
Key Concept hsm11gmse_0406_t02499.ai
Conditions for HL Theorem
To use the HL Theorem, the triangles must meet three conditions.
Conditions
• There are two right triangles.
• The triangles have congruent hypotenuses.
• There is one pair of congruent legs.
174
Lesson 4-6 Congruence in Right Triangles
Problem 1
TEKS Process Standard (1)(F)
Proof Using the HL Theorem
On the basketball backboard brackets shown below, ∠ADC and ∠BDC are
right angles and AC ≅ BC. Are △ADC and △BDC congruent? Explain.
A
How can you
visualize the two
right triangles?
Imagine cutting △ABC
along DC. On either
side of the cut, you get
triangles with the same
leg DC.
D
C
B
• You are given that ∠ADC and ∠BDC are right angles. So, △ADC and △BDC are right triangles.
• The hypotenuses of the two right triangles are AC and BC. You are given that AC ≅ BC.
• DC is a common leg of both △ADC and △BDC. DC ≅ DC by the Reflexive Property of Congruence.
Yes, △ADC ≅ △BDC by the HL Theorem.
PearsonTEXAS.com
175
Problem 2
Proof Writing a Proof Using the HL Theorem
D
B
Given: BE bisects AD at C,
AB # BC, DE # EC, AB ≅ DE
How can you get
started?
Identify the hypotenuse
of each right triangle.
Prove that the
hypotenuses are
congruent.
C
E
A
Prove: △ABC ≅ △DEC
BE bisects AD.
AC ≅ DC
Given
Def. of bisector
hsm11gmse_0406_t02545
∠ABC and
∠DEC are
right ⦞.
AB ⊥ BC
DE ⊥ EC
Given
Def. of ⊥ lines
△ ABC and △ DEC
are right .
△ABC ≅ △DEC
Def. of right triangle
HL Theorem
AB ≅ DE
NLINE
HO
ME
RK
O
Given
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
hsm11gmse_0406_t02546.ai
1.
Justify Mathematical Arguments (1)(G) Copy the flow chart and
complete the proof.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
R
S
T
Given: PS ≅ PT , ∠PRS ≅ ∠PRT
Prove: △PRS ≅ △PRT
∠PRS and ∠PRT are ≅.
Given
P
∠PRS and ∠PRT
are right ⦞.
a.
∠PRS and ∠PRT
are supplementary.
⦞ that form a linear
pair are supplementary.
PS ≅ PT
△PRS and △PRT
are right .
b.
hsm11gmse_0406_t02550.ai
c.
PR ≅ PR
△PRS ≅ △PRT
e.
d.
2.
Study Exercise 1. Can you prove that △PRS ≅ △PRT without using the HL
Theorem? Explain.
3.
Explain Mathematical Ideas (1)(G) Complete the paragraph proof. B
hsm11gmse_0406_t02551.ai
D
Given: ∠A and ∠D are right angles, AB ≅ DE
Prove: △ABE ≅ △DEB
A
Proof: It is given that ∠A and ∠D are right angles. So, a. ? by the
definition of right triangles. b. ? , because of the Reflexive Property of
Congruence. It is also given that c. ? . So, △ABE ≅ △DEB by d. ? .
176
Lesson 4-6 Congruence in Right Triangles
E
4.Given: HV # GT , GH ≅ TV ,
Proof
I is the midpoint of HV
5.Given: PM ≅ RJ ,
PT # TJ , RM # TJ ,
M is the midpoint of TJ
Proof
Prove: △IGH ≅ △ITV
Prove: △PTM ≅ △RMJ
G
P
V
I
H
T
T
J
M
R
Connect Mathematical Ideas (1)(F) For what values of x and y are the triangles
congruent by HL?
6.
hsm11gmse_0406_t02553.ai
x
x3
3y
7.
y1
3y x
hsm11gmse_0406_t02554x 5
yx
y5
8.
Apply Mathematics (1)(A) △ABC and △PQR are
right triangular sections of a fire escape, as shown.
Is each story of the building the same height?
hsm11gmse_0406_t02555
hsm11gmse_0406_t02556
Explain.
9.
Connect Mathematical Ideas (1)(F) “Aha!”
exclaims your classmate. “There must be an HA
Theorem, sort of like the HL Theorem!” Is your
classmate correct? Explain.
10.Given: △LNP is isosceles with base NP,
Proof
MN # NL, QP # PL, ML ≅ QL
C
B
A
R
Prove: △MNL ≅ △QPL
L
M
Q
N
P
P
Q
Create Representations to Communicate Mathematical Ideas (1)(E) Copy the triangle and construct a triangle congruent to it using the
givenhsm11gmse_0406_t02558
method.
11.SAS
12.HL
13.ASA
14.SSS
hsm11gmse_0406_t02559
PearsonTEXAS.com
177
15.Given: △GKE is isosceles with
Proof
base GE, ∠L and ∠D are
right angles, and K is the
midpoint of LD.
Prove: LG ≅ DE
K
L
16. Given: LO bisects ∠MLN ,
OM # LM, ON # LN
Proof
Prove: △LMO ≅ △LNO
M
O
D
L
G
N
E
17.Justify Mathematical Arguments (1)(G) Are the
triangles at the right congruent? Explain.
hsm11gmse_0406_t02560
C
F
5
hsm11gmse_0406_t02561
13
B
5
E
13
A
Analyze Mathematical Relationships (1)(F) For Exercises 18 and 19,
use the figure at the right.
18.Given: BE # EA, BE # EC, △ABC is equilateral
Proof
Prove: △AEB ≅ △CEB
D
B
hsm11gmse_0406_t10988.ai
E
A
19.Given: △AEB ≅ △CEB, BE # EA, BE # EC
C
Can you prove that △ABC is equilateral? Explain.
hsm11gmse_0406_t02562
TEXAS Test Practice
20.You often walk your dog around the
neighborhood. Based on the diagram
at the right, which one of the following statements about distances is true?
A.SH = LH C.SH 7 LH
B.PH = CH
D.PH 6 CH
School (S)
Park (P)
Home (H)
Café (C )
Library (L)
X
21.In equilateral △XYZ, name four pairs of congruent right
triangles. Explain why they are congruent.
hsm11gmse_0406_t02563
Q
P
Y
178
Lesson 4-6 Congruence in Right Triangles
S
R
Z
hsm11gmse_0406_t02564
4-7 Congruence in Overlapping Triangles
TEKS FOCUS
VOCABULARY
•Representation – a way to display or describe information.
TEKS (6)(B) Prove two triangles are congruent by
applying the Side-Angle-Side, Angle-Side-Angle, SideSide-Side, Angle-Angle-Side, and Hypotenuse-Leg
congruence conditions.
You can use a representation to present mathematical
ideas and data.
TEKS (1)(E) Create and use representations to organize,
record, and communicate mathematical ideas.
Additional TEKS (1)(F), (1)(G)
ESSENTIAL UNDERSTANDING
You can sometimes use the congruent corresponding parts of one pair of congruent
triangles to prove another pair of triangles congruent. This often involves overlapping
triangles.
Problem 1
Identifying Common Parts
What common angle do △ACD and △ECB share?
E
A
B
How can you see an
individual triangle in
order to redraw it?
Use your finger to
trace along the lines
connecting the three
vertices. Then cover up
any untraced lines.
D
C
Separate and redraw △ACD and △ECB.
A
E
D
B
hsm11gmse_0407_t05049.ai
C
C
The common angle is ∠C.
hsm11gmse_0407_t05050.ai
PearsonTEXAS.com
179
Problem 2
TEKS Process Standard (1)(E)
Proof Using Common Parts
Z
Y
Given: ∠ZXW ≅ ∠YWX, ∠ZWX ≅ ∠YXW
Prove: ZW ≅ YX
W
• ∠ZXW ≅ ∠YWX and ∠ZWX ≅ ∠YXW
• The diagram shows that △ZWX and △YXW are
overlapping triangles.
hsm11gmse_0407_t05052
Show △ZWX ≅ △YXW . Then use
corresponding parts of congruent
triangles to prove ZW ≅ YX.
A diagram of the triangles separated
Z
W
Y
XW
X
X
∠ZXW ≅ ∠YWX
Given
hsm11gmse_0407_t05053
WX ≅ WX
△ZWX ≅ △YXW
ZW ≅ YX
Reflexive Prop. of ≅
ASA
Corresp. parts of
≅ are ≅.
∠ZWX ≅ ∠YXW
Given
Problem 3
Proof Using Two Pairs of Triangles
hsm11gmse_0407_t05054
How do you choose
another pair of
triangles to help in
your proof?
Look for triangles that
share parts with △GED
and △JEB and that you
can prove congruent.
In this case, first prove
△AED ≅ △CEB.
Given: In the origami design, E is the midpoint of AC and DB.
Prove: △GED ≅ △JEB
Proof: E is the midpoint of AC and DB, so AE ≅ CE
and DE ≅ BE. ∠AED ≅ ∠CEB because vertical
angles are congruent. Therefore, △AED ≅ △CEB
by SAS. ∠D ≅ ∠B because corresponding
parts of congruent triangles are congruent.
∠GED ≅ ∠JEB because vertical angles are
congruent. Therefore, △GED ≅ △JEB by ASA.
A
G
D
180
Lesson 4-7 Congruence in Overlapping Triangles
B
E
J
C
Problem 4
TEKS Process Standard (1)(G)
Proof Separating Overlapping Triangles
C
Given: CA ≅ CE , BA ≅ DE
Prove: BX ≅ DX
Which triangles are
useful here?
If △BXA ≅ △DXE,
then BX ≅ DX
because they are
corresponding parts. If
△BAE ≅ △DEA,
you will have enough
information to show
△BXA ≅ △DXE.
B
D
X
E
A
B
D
X
A
B
E
D
A
Statements
E
E
A
Reasons
1) BA ≅ DE
1) Given
2) CA ≅ CE
2) Given
3) ∠CAE ≅ ∠CEA
3) Base ⦞ of an isosceles △ are ≅.
4) AE ≅ AE
4) Reflexive Property of ≅
5) △BAE ≅ △DEA
5) SAS
6) ∠ABE ≅ ∠EDA
s are ≅.
6) Corresp. parts of ≅ △
7) ∠BXA ≅ ∠DXE
7) Vertical angles are ≅.
8) △BXA ≅ △DXE
8) AAS
9) BX ≅ DX
s are ≅.
9) Corresp. parts of ≅ △
NLINE
HO
ME
RK
O
hsm11gmse_0407_t05057
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
In each diagram, the red and blue triangles are congruent. Identify their common
side or angle.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
K
2.
P
L
E 3. X
D
N
T
W
G
F
Z
Y
M
Separate and redraw the indicated triangles. Identify any common sides or angles.
4.
△PQS and △QPR
5.△ACB
and △PRB
6.△JKL and △MLK
hsm11gmse_0407_t02768
P
A
Q
hsm11gmse_0407_t02767
P
hsm11gmse_0407_t02769
K
L
O
C
R
T
S
B
J
M
R
PearsonTEXAS.com 181
hsm11gmse_0407_t02772
hsm11gmse_0407_t02771
hsm11gmse_0407_t02770
7.
Justify Mathematical Arguments (1)(G) Complete the flow proof.
P
Given: ∠T ≅ ∠R, PQ ≅ PV
Prove: ∠PQT ≅ ∠PVR
Q
V
∠T ≅ ∠R
a.
S
R
T
∠TPQ ≅ ∠RPV
△TPQ ≅ △RPV
b.
∠PQT ≅ ∠PVR
e.
d.
hsm11gmse_0407_t02773
PQ ≅ PV
c.
8.Given: RS ≅ UT , RT ≅ US
9.Given: QD ≅ UA, ∠QDA ≅ ∠UAD
Prove: △RST ≅ △UTS
S
T
Prove: △QDA ≅ △UAD
U
Q
Proof
hsm11gmse_0407_t02774
M
R
D
V
10.Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4
Proof
Prove: △QET ≅ △QEU
T
hsm11gmse_0407_t02775
3
4
R
U
W
Q
Proof
1
2
E
11.Given: AD ≅ ED, D is the midpoint of BF
Proof
Prove: △ADC ≅ △EDG
hsm11gmse_0407_t02776
A
G
B
U
A
F
B
D
E
C
12.Explain Mathematical Ideas (1)(G) In the diagram at the right,
∠V ≅ ∠S, VU ≅ ST, and PS ≅ QV. Which two triangles are
congruent
by SAS? Explain.
hsm11gmse_0407_t02777.ai
Q
P
X
W
hsm11gmse_0407_t02778.ai
V
13.Identify a pair of overlapping congruent triangles in the
diagram. Then use the given information to write a proof
to show that the triangles are congruent.
Given: AC ≅ BC, ∠A ≅ ∠B
U
R
S
T
A
B
F
E
hsm11gmse_0407_t02779.ai
D
C
hsm11gmse_0407_t02784
182
Lesson 4-7 Congruence in Overlapping Triangles
Mathematics (1)(A) The figure at the right is
part of a clothing design pattern, and it has the
following relationships.
STEM14.Apply
G
E
B
J
4 H 8 9
I
• GC # AC
• AB # BC
• AB } DE } FG
• m∠A = 50
A
D
1
F
2
7
5
3
6
C
• △DEC is isosceles with base DC.
a.Find the measures of all the numbered angles in the figure.
b.Suppose AB ≅ FC. Name two congruent triangles and explain how you can
prove them congruent.
15.Given: AC ≅ EC , CB ≅ CD
16.Given: QT # PR, QT bisects PR,
Proof
QT bisects ∠VQS
Q
Prove: VQ ≅ SQ
P
Proof
Prove: ∠A ≅ ∠E
C
B
A
D
F
V
E
R
S
T
17.Create Representations to Communicate Mathematical Ideas (1)(E) Draw a
Proof quadrilateral ABCD with AB } DC, AD } BC, and diagonals AC and DB intersecting
at E. Label your diagram to indicate the parallel sides.
hsm11gmse_0407_t02780.ai
a.List all the pairs of congruent segments in your diagram.
hsm11gmse_0407_t02781.ai
b.Explain how you know that the segments you listed are congruent.
TEXAS Test Practice
18.According to the diagram at the right, which statement is true?
A.△DEH ≅ △GFH by AAS
C.△DEF ≅ △GFE by AAS
B.△DEH ≅ △GFH by SAS
D.△DEF ≅ △GFE by SAS
19.△ABC is isosceles with base AC. If m∠C = 37, what is m∠B?
F.
37
G.74
H.106
J.143
G
F
H
E
20.Which word correctly completes the statement “All ? angles are
D
congruent”?
A.adjacent
B.supplementary C.right
D.corresponding
J
G
21.In the figure, LJ } GK and M is the midpoint of LG.
a.Copy the diagram. Then mark your diagram with the given information.
hsm11gmse_0407_t05015
b.Prove △LJM ≅ △GKM.
M
L
c.Can you prove that △LJM ≅ △GKM another way? Explain.
K
PearsonTEXAS.com
183
hsm11gmse_0407_t05018
Topic 4 Review
TOPIC VOCABULARY
• base of an isosceles
triangle, p. 168
• base angles of an isosceles
• congruent polygons, p. 148
• hypotenuse, p. 174
• legs of an isosceles triangle, p. 168
• legs of a right triangle, • corollary, p. 168
• vertex angle of an p. 174
triangle, p. 168
isosceles triangle, p. 168
Check Your Understanding
Choose the correct term to complete each sentence.
1. The two congruent sides of an isosceles triangle are the
2. The side opposite the right angle of a right triangle is the
3. A
4. ?
?
? .
? .
to a theorem is a statement that follows immediately from the theorem.
have congruent corresponding parts.
4-1 Congruent Figures
Quick Review
Exercises
Congruent polygons have congruent corresponding sides
and angles. When you name congruent polygons, always list
corresponding vertices in the same order.
RSTUV @ KLMNO. Complete the congruence
statements.
Example
HIJK @ PQRS. Write all possible congruence statements.
The order of the parts in the congruence statement tells you
which parts correspond.
Sides: HI ≅ PQ, IJ ≅ QR, JK ≅ RS, KH ≅ SP
5.TS ≅
6.∠N ≅
? 7.LM ≅
8.VUTSR ≅
? W
Z
80
145
X
3 Y
R
Q
100
8.6
S
35
10
5
P
9.m∠P
10.QR
11.WX
12. m∠Z
13.m∠X 14.m∠R
hsm11gmse_04cr_t05080
Topic 4 Review
?
WXYZ ≅ PQRS. Find each measure or length.
Angles: ∠H ≅ ∠P, ∠I ≅ ∠Q, ∠J ≅ ∠R, ∠K ≅ ∠S
184
?
4-2 and 4-3 Triangle Congruence by SSS, SAS, ASA, and AAS
Quick Review
Exercises
You can prove triangles congruent with limited information
about their congruent sides and angles.
15. In △HFD, what angle is included between DH
and DF ?
Postulate or Theorem
You need
Side-Side-Side (SSS)
three sides
16. In △OMR, what side is included between
∠M and ∠R?
Side-Angle-Side (SAS)two sides and an
included angle
Angle-Side-Angle (ASA)two angles and an
included side
Angle-Angle-Side (AAS)two angles and a
nonincluded side
Which postulate or theorem, if any, could you use to
prove the two triangles congruent? If there is not
enough information to prove the triangles congruent,
write not enough information.
17. 18.
19. 20.
Example
What postulate would you use to prove the triangles
congruent?
You know that three sides are
congruent. Use SSS.
hsm11gmse_04cr_t05082hsm11gmse_04cr_t05083
4-4 Using Corresponding Parts of Congruent Triangles
hsm11gmse_04cr_t05084
hsm11gmse_04cr_t05081
hsm11gmse_04cr_t05085
Exercises
Quick Review
Once you know that triangles are congruent, you can make
conclusions about corresponding sides and angles because,
by definition, corresponding parts of congruent triangles
are congruent. You can use congruent triangles in the
proofs of many theorems.
How can you use congruent triangles to prove the
statement true?
21. TV ≅ YW 22.BE ≅ DE
V
C
W
B
Example
How can you use congruent triangles to prove jQ @ jD?
W
K
D
T
E
X
Y
23. ∠B ≅ ∠D
D
24.KN ≅ ML
C
K
L
Q
E
V
Since △QWE ≅ △DVK by AAS, you know that ∠Q ≅ ∠D
because corresponding parts of congruent triangles
are congruent.
hsm11gmse_04cr_t05087hsm11gmse_04cr_t05088
B
E
D
N
M
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4-5 Isosceles and Equilateral Triangles
Quick Review
Exercises
If two sides of a triangle are congruent, then the
angles opposite those sides are also congruent by
the Isosceles Triangle Theorem. If two angles of a
triangle are congruent, then the sides opposite those
angles are congruent by the Converse of the Isosceles
Triangle Theorem.
Algebra Find the values of x and y.
Equilateral triangles are also equiangular.
25. 26.
x
50
4
x
y
27. 28.
25
y
Example
y
y
125
x
hsm11gmse_04cr_t05093
hsm11gmse_04cr_t05092
x
What is mjG?
25
7
E
Since EF ≅ EG, ∠F ≅ ∠G by the
Isosceles Triangle Theorem. So
m∠G = 30.
F
30
G
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4-6 Congruence in Right Triangles
hsm11gmse_04cr_t05091
Exercises
Quick Review
If the hypotenuse and a leg of one right triangle are
congruent to the hypotenuse and a leg of another
right triangle, then the triangles are congruent by the
Hypotenuse-Leg (HL) Theorem.
Write a proof for each of the following.
29. Given: LN # KM, KL ≅ ML
Prove: △KLN ≅ △MLN
L
Example
Which two triangles are congruent? Explain.
M
B
N
A
C
Z
L
Y
X
Since △ABC and △XYZ are right triangles with congruent
legs, and BC ≅ YZ, △ABC ≅ △XYZ by HL.
K
N
M
P
30. Given: PS # SQ, RQ # QS,
PQ ≅ RS
Q
hsm11gmse_04cr_t05097
Prove: △PSQ
≅ △RQS
S
R
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Topic 4 Review
4-7 Congruence in Overlapping Triangles
Quick Review
Exercises
To prove overlapping triangles congruent, you look for the
common or shared sides and angles.
Name a pair of overlapping congruent triangles in each
diagram. State whether the triangles are congruent by
SSS, SAS, ASA, AAS, or HL.
Example
31. E
Separate and redraw the
overlapping triangles.
Label the vertices.
F
C
E
F
E
D
33. P
hsm11gmse_04cr_t05100
C
G
I
H
B
Q
A
D
32. F
F
D
E
A
R
S
hsm11gmse_04cr_t05102
T
hsm11gmse_04cr_t05099 hsm11gmse_04cr_t05101
C
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Topic 4 TEKS Cumulative Practice
Multiple Choice
4.Given: ∠1 ≅ ∠2, AB ≅ AC
Read each question. Then write the letter of the correct
answer on your paper.
A
1.Given: DE } CB,
∠ADE ≅ ∠AED
What additional information do you need to prove
△ABD ≅ △ACE by AAS?
A
Prove: AC ≅ AB
E
D
Proof: Since DE } CB,
4 62
15 3
∠ACB ≅ ∠ADE and
C
B
E
D
∠AED ≅ ∠ABC by the
B
C
Corresponding Angles
F.
AD ≅ AEH.
∠5 ≅ ∠6
Theorem. Since ∠ADE ≅ ∠AED,
G.
BD ≅ ECJ.
BE ≅ DC
∠ACB ≅ ∠ABC by the Transitive Property
of Congruence. Which theorem or definition
5.Which of the
following statements is true?
hsm11gmse_04cu_t05071
proves that AC ≅ AB?
hsm11gmse_04cu_t05070
A.
Point, line, and plane are undefined terms.
A.
Isosceles Triangle Theorem
B.
A theorem is an accepted statement of fact.
B.
Converse of Isosceles Triangle Theorem
C.
“Vertical angles are congruent” is a definition.
C.
Alternate Interior Angles Theorem
D.
A postulate is a conjecture that is proven.
D.
Definition of congruent segments
6.Which condition allows you to prove that / } m?
2.Which statement must be true for two polygons to be
congruent?
1 8
F.
All the corresponding sides should be congruent.
2 3
G.
All the corresponding sides and angles should
be congruent.
4
5
m
H.
All the corresponding angles should be congruent.
J.
All sides in each polygon should be congruent.
3.If △ABC ≅ △CDA, which of the following must
be true?
A
B
F.
∠1 ≅ ∠8H.
∠3 ≅ ∠4
G.
∠2 ≅ ∠8J.
∠3 ≅ ∠5
7.hsm11gmse_04cu_t05072
The measure of one base angle of an isosceles
triangle is 23. What is the measure of the vertex angle?
A.
113C.
23
B.
134D.
67
D
C
A.
AB ≅ CAC.
∠CAB ≅ ∠ACD
B.
BC ≅ DCD.
∠ABC ≅ ∠CAD
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Topic 4 TEKS Cumulative Practice
Gridded Response
13.Write a proof for the following.
8.What is the value of x in the figure below?
D
A
S
Prove: △GAT ≅ △TSG
75
T
L
14.Write a proof for the following.
Given: LN bisects ∠OLM
and ∠ONM.
(5x 6)
44
E
Given: AT ≅ SG,
AT } GS
G
F
9.What is the value of x in the figure below?
Prove: ON ≅ MN
hsm11gmse_04ct_t02583
M
O
N
(2x 5)
15. Isosceles △ABC, with right ∠B, has a point D on AC
such that BD # AC. What is the relationship between
△ABD and △CBD? Explain.
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hsm11gmse_04ct_t02584
10. ABCD ≅ WXYZ. What is WX?
A
3
Y
B
2
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4
D
C
X
W
Constructed Response
11. Is GK ≅ HK ? Explain.
hsm11gmse_04cu_t05076
G
H
K
hsm11gmse_04cu_t05079
b.How confident are you about your conjecture?
Explain.
J
12. Write a proof for the following.
Given: AE ≅ DE, EB ≅ EC
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Prove: △AEB ≅ △DEC
A
Halley’s Comet can be seen periodically at its perihelion, the
shortest distance from the sun during its orbit. Mark Twain was
born two weeks after the comet’s perihelion. In his
biography he said, “I came in with Halley’s Comet in 1835. It is
coming again next year, and I expect to go out with it.” Twain
died in 1910, the day after the comet’s perihelion. The most
recent sighting of Halley’s Comet was in 1986. Its next
appearance is expected in 2061.
a.Make a conjecture about the year in which Halley’s
Comet will appear after 2061. Explain your
reasoning.
L
D
E
B
Z
2
4
16. Read this excerpt from an online news article.
17. The coordinates of the vertices of rectangle LMNK
are L( -2, 5), M(2, 5), N(2, 3), and K( -2, 3). The
coordinates of the vertices of rectangle PQRS are
P(3, 0), Q(3, -3), R(1, -3), and S(1, 0). Are these two
rectangles congruent? Explain why or why not. If
not, how could you change the vertices of one of the
rectangles to make them congruent?
C
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