Download 4.2 Applying Congruence

Geometry
Name ______________________
Period ___________
Date ___________
Summary of Chapter 4: Congruent Triangles
4.2 Applying Congruence
1. Two figures are congruent if they have the same Size and Shape.
2. What is a congruence statement? What can it help us do?
It helps us to identify corresponding parts.
3. In the diagram, EFG  OPQ. Complete the statement.
13 cm
a)
b)
c)
d)
e)
f)
g)
EF  OP
P  F
G  Q
mO = 110⁰
QO = 7 cm
GFE  GPO
x = 10
110o
7 cm
(3x-10)o
50o
4. What is the third angles theorem?
If two angles of one triangle are congruent to two angles of another triangle, then the third
angles are also congruent.
5. Given  ABC  DEF . Find the values of the missing variables.
X = 17
Y = 17
6. Label the two triangles with the given information and solve for x, y, and z. Show all work.
Given : APT WOC.
m∠A = 30°
x = 120
m∠P = 100°
m∠W = (y + 20)°
y = 10
m∠O = (x – 20)°
z = 20
m∠C = (z +30)°
4.3-4.5 SSS, SAS, HL, ASA & AAS
7. Draw two triangles and label them such that the SSS Postulate would prove them congruent. Write a
congruence statement based on your diagram.
ABC  XYZ
8. Draw two triangles and label them such that the SAS Congruence Postulate would prove them congruent.
Write a congruence statement based on your diagram.
IJH  KML
9. Draw two triangles and label them such that the Hypotenuse – Leg would prove them congruent. Write a
congruence statement based on your diagram.
ABC  DEF
10. Draw two triangles and label them such that the ASA Postulate would prove them congruent. Write a
congruence statement based on your diagram.
ABC  XYZ
11. Draw two triangles and label them such that the AAS Congruence Theorem would prove them congruent.
Write a congruence statement based on your diagram.
ABC  XYZ
12. Why doesn’t SSA work?
Check this out, http://www.mathopenref.com/congruentssa.html
State the third congruence that must be given to prove that JRM  DFB using the indicated postulate or
theorem.
13. . GIVEN: JR  DF , RM  FB
MJ  BD Use the SSS Congruence Postulate.
14. GIVEN: JR  DF , RM  FB
R  F Use the SAS Congruence Postulate.
15. GIVEN: JM  DB J is a right angle and J  D RM  FB Use the HL Theorem.
16. GIVEN:
R  F , RM  FB
17. GIVEN:
R  F , RM  FB
M  B Use the ASA Congruence Postulate.
J  D Use the AAS Congruence Theorem.
In the figures below mark any additional angles or sides that you know are congruent just from the given
information. If you CANNOT mark any additional angles or sides, write “Figure Complete”.
18.
SSS
21.
HL
P
19.
22. NEI
V
S
SAS
20. AAS
23. AAS
H
For #24 – 26, decide if the following triangles are congruent. For each problem, do the following:
MARK YOUR FIGURES!
a) Can these triangles be proven congruent? Write “Yes” or “No”.
b) Name the congruent triangles (ONLY if they are congruent) and
c) State the postulate or theorem that supports your answer ((ONLY if they are congruent)
24.
25.
A
26.
B
C
a) YES
a) NO
a) YES
b) ABE  DBC
b) ________  ________
b) ABC  ADC
c) AAS
c) ___________________
c) HL
D
27. Prove that
QRS  TUS
Given: S is a midpoint of
QU
and
RT
Statements
1. S is a midpoint of
2.
3.
4.
5.
QU
and
Reasons
RT
Given
QS  US
Definition of midpoint
RS  TS
QSR  UST
QRS  TUS
3. Def. of midpoint
4. Vertical Angles Congruence Postulate
SAS Congruence Postulate
DEF  HGF
DH  GE , F is a midpoint of GE , and DE  GH
28. Prove that
Given:
Statements
Reasons
DH  GE
GFH is right angle
Given
Def. of Perpendicular Lines
5.
EFD is right angle
DEF is right triangle
HGF is right triangle
F is a midpoint of GE
6.
GF  EF
Def. of Midpoint
7.
8.
DE  GH
DEF  HGF
Given
1.
2.
3.
4.
Def. of Perpendicular Lines
Definition of Right Triangle.
* CHANGED
Given
HL Congruence Theorem
4.6 CPCTC
29. What does CPCTC mean?
Corresponding Parts of Congruent Triangles are Congruent.
AD  BC
AB DC and AD BC
30. Prove that
Given:
Statements
Reasons
1.
2.
AB DC
DCA  BAC
Given
3.
4.
AD BC
DAC  BCA
Given
5. Reflexive Property of Congruence
6.
AC AC
ADC  CBA
7.
AD  BC
CPCTC
5.
2. Alternate Interior Angles Theorem
4. Alternate Interior Angles Theorem
ASA Congruence Postulate
31. Write an explanation IN COMPLETE SENTENCES of how you can prove the statement is true. Make sure to
justify each piece of information with a valid reasoning. Mark your pictures as you go. Remember to use one
of the five postulates/theorems (SSS, SAS, ASA, AAS, HL).
Prove that JT  JR
By given, we have
TM  RM .
We also know that JM  JM by reflexive property. So far we have two congruent
sides.
Because TMR is a straight angle (180⁰) and JMR is 90⁰, (Definition of right
angle) JMT is also 90⁰ by angle addition postulate. ( JMR and JMT are linear
pairs.) We know JMR  JMT because they are both 90⁰. (Def. of congruent
angles) They are both right triangles but we can’t use HL here because we don’t know if
JT  JR . But we do know that JMR and JMT are congruent included angles,
so by SAS Congruence Postulate, JMR  JMT . BY CPCTC, we have JT  JR