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Name———————————————————————— Lesson
4.5
Date —————————————
Study Guide
For use with the lesson “Prove Triangles Congruent by SAS and HL”
Use sides and angles to prove congruence.
Lesson 4.5
goal
Vocabulary
In a right triangle, the sides adjacent to the right angle are called
the legs.
The side opposite the right angle is called the hypotenuse of the
right triangle.
Postulate 20 Side-Angle-Side (SAS) Congruence Postulate: 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.
Theorem 5 Hypotenuse-Leg Congruence Theorem: If the
hypotenuse and a leg of a right triangle are congruent to the
hypotenuse and a leg of a second right triangle, then the two triangles
are congruent.
example 1
Use the SAS Congruence Postulate
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Prove that nABC > nDEF.
B
F
D
C
A
E
Solution
} } } }
​  > EF​
​  , and ∠ B > ∠ E.
The marks on the diagram show that AB​
​  > DE​
​  , BC​
So, by the SAS Congruence Postulate, n ABC > n DEF.
Exercises for Example 1
Decide whether enough information is given to prove that the triangles
are congruent using the SAS Congruence Postulate.
1. n PQT, n RQS
P
2. n NKJ, nLKM
R
J
3. nWXY, n ZXY
X
Z
L
K
Q
T
S
W
N
Y
M
Geometry
Chapter Resource Book
4-67
Name———————————————————————— Lesson
Lesson 4.5
4.5
Date —————————————
Study Guide continued
For use with the lesson “Prove Triangles Congruent by SAS and HL”
example 2
Use the Hypotenuse-Leg Theorem
Write a proof.
} } } } } }
Given: AB​
​  > DC​
​  , BA​
​  ⊥ ​AC​,  ​CD​ ⊥ DB​
​  
A
C
B
D
Prove: n ABC > n DCB
Solution
B
Redraw the triangles so they are side by side with the corresponding parts in the same
position. Mark the given information in
the diagram.
C
A
C
B
D
Exercises for Example 2
Write a proof.
}
} } }
4. Given: ​AB​ > DB​
​  , BC​
​  ⊥ AD​
​  
5. Given: m∠ JKL 5 m∠ MLK 5 908
} }
Prove: n ABC > n DBC​JL​ > MK​
​  
Prove:n JKL > n MLK
B
A
4-68
Geometry
Chapter Resource Book
C
K
J
L
M
D
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
StatementsReasons
} } } }
​  
1. Given
1. ​BA​ ⊥ ​AC​,  ​CD​ ⊥ DB​
2. Definition of ⊥ lines
2. ∠ A and ∠ D are right angles.
3. n ABC and n DCB are right triangles. 3. Definition of a right triangle
} }
​  
4. Reflexive Property of Congruence
H 4. ​CB​ > BC​
} }
​  
5. Given
L 5. ​AB​ > DC​
6. HL Congruence Theorem
6. n ABC > n DCB
Lesson 4.5 Prove Triangles
Congruent by SAS and HL,
­continued
answers
}
5.
Statements
}
12. not enough 13. RM​
​  ù FB​
​   14. ∠ J ù ∠ D
}
} } } }
} }
​  ; Vertical Angles Theorem;
Given; ​CB​ ù BD​
}
15. JM​
​  ù DB​
​  or JR​
​  ù DF​
​   16. Given; ​AB​ ù BE​
​  ;
SAS Congruence Postulate 17. Given; Alternate
Interior Angles Theorem; Given; Reflexive
Property of Congruence; SAS Congruence
Postulate
Practice Level C
1. not enough 2. enough; HL 3. not enough
} }
4. enough; SAS 5. ∠ BCA; ∠ EDF 6. BC​
​  ; ED​
​  
} }
7. ​AC​;  ​FD​  8. They are congruent by SAS.
9. Definition of perpendicular lines; n PRS and
n QSR are right triangles; Reflexive Property of
Congruence; HL Congruence Theorem
10. ∠ OML and ∠ OMN are right angles;
∠ OML ù ∠ OMN; Given; Reflexive Property of
Congruence; SAS Congruence Postulate
Reasons
1. ∠ JKL and ∠ MLK 1. Given
are right angles.
2. n JKL and n MLK 2. Def. of a right
are right triangles. triangle
} }
​  
3. Given
3. ​JL​ > MK​
} }
​  
4. Reflexive Property
4. ​KL​ > LK​
of Congruence
5. n JKL > n MLK
5. HL Congruence
Theorem
Problem Solving Workshop:
Mixed Problem Solving
}
}
1. a. The legs are ​MN​ and NL​
​  . The hypotenuse is ​
}
ML​ . b. 45° to 60°
y
2. a.
C
4
B
A
x
2
D
Study Guide
E
1. Yes; You are given that two sides and the
4.
Reasons
Statements
} }
H 1. ​AB​ > DB​
​  
1. Given
} }
​  
2. Given
2. ​BC​ ⊥ AD​
3. ∠ ACB and ∠ DCB 3. Def. of ⊥ lines
are right angles.
4. n ABC and n DCB 4. Def. of a right
are right triangles. triangle
} }
​  
5. Reflexive Property
L 5. ​BC​ > BC​
of Congruence
6. n ABC > n DBC 6. HL Congruence
Theorem
A52
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C4AK.indd 52
F
}
b. n ABC is a right triangle because AB​
​  has a
}
1
slope of }
​ 2 ​and BC​
​   has a slope of 22. nDEF is
1
}
a right triangle because DE​
​  has a slope of }
​ 2 ​and
}
}
}
​EF​ has a slope of 22. c. The legs ​AB​ and DE​
​  
}
​
both have a length of 3​Ï5 ​ and the hypotenuses
}
}
}
AC​ and ​DF​ both have a length of 3​Ï10 ​.  So, by the
Hypotenuse-Leg Congruence Theorem,
n ABC > nDEF.
}
}
}
3. a. 2​Ï3x ​ 1 5​Ï3x ​ 1 3​Ï3x ​ 5 180 b. 36°, 90°,
54° c. right triangle 4. Yes; because the flag is a
rectangle, you know that ∠ PJL, ∠ JLM, ∠ LMP, and
} } } }
​ . JL​
​  i PM​
​  and ​
∠ MPJ are right angles and JP​
​  > LM​
} }
JP​ i LM​
​  by the Lines Perpendicular to a Transversal
Theorem. Using the Alternate Interior Angles Theorem, you know that ∠ JKP > ∠ MNL. So, nJKP >
nMNL by the AAS Congruence Theorem. 5. 458
6. Answers will vary.
7. a.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
i­ncluded angle of one triangle are congruent to two
sides and the included angle of another ­triangle.
2. Yes; ∠ JKN and ∠ MKL are congruent because
they are vertical angles. So you have two sides
and the included angle of one triangle that are
congruent to two sides and the included angle of
another triangle. 3. No; You have two sides in
nWXY that are congruent to two sides in n ZXY,
but the angle in n ZXY is not the included angle.
Reasons
Statements
} }
​  
1. Given
1. ​BD​ > AC​
} }
​  , 2. Given
2. ​AB​ ⊥ BC​
} }
​  
​DC​ ⊥ BC​
4/28/11 6:14:09 PM