Lesson 7.1
Right Triangles
pp. 262-266
Objectives:
1. To prove special congruence
theorems for right triangles.
2. To apply right triangle congruence
theorems in other proofs.
Review
C
ABC is a rt. 
B is the rt. 
A
B
The side opposite B is AC,
called the hypotenuse.
AB and BC are called the legs.
SAS
ASA
AAS
SSS
Theorem 7.1
HL Congruence Theorem. If the
hypotenuse and a leg of one
right triangle are congruent to
the hypotenuse and
corresponding leg of another
right triangle, then the two
triangles are congruent.
HL Congruence Theorem
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HL Congruence Theorem
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Theorem 7.2
LL Congruence Theorem. If the
two legs of one right triangle
are congruent to the two legs
of another right triangle, then
the two triangles are
congruent.
LL Congruence Theorem
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Theorem 7.3
HA Congruence Theorem. If
the hypotenuse and an acute
angle of one right triangle are
congruent to the hypotenuse
and corresponding acute angle
of another right triangle, then
the two triangles are
congruent.
HA Congruence Theorem
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Theorem 7.4
LA Congruence Theorem. If a
leg and one of the acute angles
of a right triangle are
congruent to the
corresponding leg and acute
angle of another right triangle,
then the two triangles are
congruent.
LA Congruence Theorem
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For the next 5 questions
decide whether the right
triangles are congruent. If they
are, identify the theorem that
justifies it. Be prepared to give
the congruence statement.
Practice: Is ∆ADC  ∆ABC?
B
1. HL
2. LL
C
3. HA
4. LA
5. Not
D
enough A
information
Practice: Is ∆EFG  ∆EHG?
F
1. HL
2. LL
G
3. HA
4. LA
5. Not
H
enough E
information
Practice: Is ∆LMN  ∆PQR?
Q
P
1. HL
N
2. LL
3. HA
4. LA
R
5. Not
L
M
enough
information
Practice: Is ∆XYZ  ∆YXW?
1. HL
W
Z
2. LL
3. HA
4. LA
5. Not
X
Y
enough
information
Practice: Is ∆LMO  ∆PNO?
1. HL
M
P
2. LL
3. HA
O
4. LA
N
5. Not L
enough
information
Homework
pp. 264-266
►A. Exercises
Identify any triangle congruence theorem
or postulate corresponding to each right
triangle congruence theorem below.
(Remember that the right angles are
congruent.)
3. LA, adjacent case
►A. Exercises
Identify any triangle congruence theorem
or postulate corresponding to each right
triangle congruence theorem below.
(Remember that the right angles are
congruent.)
4. LA, opposite case
►A. Exercises
Identify any triangle congruence theorem
or postulate corresponding to each right
triangle congruence theorem below.
(Remember that the right angles are
congruent.)
5. HL
►A. Exercises
9. Use the diagram to state a triangle
congruence. Which right triangle
theorem justifies the statement?
M
H
P
N
Q
►A. Exercises
10. Prove HA.
T
R
S
W
U
V
10.
Statements
1. RST & UVW
are rt. ’s; RT 
UW; R U
2. S & V are rt.
’s
3. S  V
4. RST  UVW
Reasons
1. Given
2. Def. of rt. ’s
3. All rt. ’s are 
4. SAA
►B. Exercises
Use the same diagram as in exercise 10
for the proofs in exercises 11-12 (the two
cases of the LA Congruence Theorem).
11. LA (opposite case)
Given: ∆RST and ∆UVW are right
triangles; RS  UV;
T  W
Prove: ∆RST  ∆UVW
►B. Exercises
11. LA (opposite case)
Given: ∆RST and ∆UVW are right
triangles; RS  UV;
T  W
Prove: ∆RST  ∆UVW
R
T
W
S U
V
►B. Exercises
12. LA (adjacent case)
Given: ∆RST and ∆UVW are right
triangles; RS  UV;
R  U
Prove: ∆RST  ∆UVW
R
T
W
S U
V
12.
Statements
1. RST & UVW
are rt. ’s; RS 
UV; R U
2. S & V are rt.
’s
3. S  V
4. RST  UVW
Reasons
1. Given
2. Def. of rt. ’s
3. All rt. ’s are 
4. ASA
►B. Exercises
Use the following diagram to prove
exercise 13.
13. Given: P and Q are right angles;
PR  QR
Q
Prove: PT  QT
R
T
P
►B. Exercises
Use the following diagram to prove
exercises 15-19.
15. Given: WY  XZ; X  Z
Prove: ∆XYW  ∆ZYW
W
X
Y
Z
■ Cumulative Review
Give the measure of the angle(s) formed by
22. two opposite rays.
■ Cumulative Review
Give the measure of the angle(s) formed by
23. perpendicular lines.
■ Cumulative Review
Give the measure of the angle(s) formed by
24. an equiangular triangle.
■ Cumulative Review
Give the measure of the angle(s) formed by
25. the bisector of a right angle.
■ Cumulative Review
26. Which symbol does not represent a
set?
ABC, ∆ABC, A-B-C, {A, B, C}