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 ___________________ and _______________________.
2. What is a congruence statement? What can it help us do?
3. In the diagram, EFG
OPQ. Complete the statement.
13 cm
a) EF
_____
b)
P
_____
c)
G
_____
d) m O = _____
e) QO = _____
f)
GFE _____
g)
x = _____
110o
7 cm
(3x-10)o
50o
4. What is the third angles theorem?
5. Given
ABC
DEF . Find the values of the missing variables.
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 = ______
m P = 100°
m W = (y + 20)°
y = ______
m O = (x – 20)°
z = ______
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.
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.
9. Draw two triangles and label them such that the Hypotenuse – Leg would prove them congruent. Write a
congruence statement based on your diagram.
10. Draw two triangles and label them such that the ASA Postulate would prove them congruent. Write a
congruence statement based on your diagram.
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.
12. Why doesn’t SSA work?
State the third congruence that must be given to prove that JRM
theorem.
13. . GIVEN: JR
14. GIVEN: JR
15. GIVEN: JM
16. GIVEN:
R
17. GIVEN:
R
DF , RM
DF , RM
FB
FB
____ ____ Use the SSS Congruence Postulate.
____ ____ Use the SAS Congruence Postulate.
DB J is a right angle and
F , RM
F , RM
FB
FB
DFB using the indicated postulate or
J
D ____ ____Use the HL Theorem.
____ ____ Use the ASA Congruence Postulate.
____ ____ 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..
21.
V
P
S
19.
20.
22.
23.
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
D
a) ___________________
a) ___________________
a) ___________________
b) ________
b) ________
b) ________
________
c) ___________________
________
c) ___________________
________
c) ___________________
27. Prove that
QRS
TUS
Given: S is a midpoint of
QU and RT
Statements
28. Prove that
Given:
DH
Reasons
DEF HGF
GE , F is a midpoint of GE , and DE GH
Statements
DH
GE , F is a midpoint of GE , and DE
Reasons
GH
4.6 CPCTC
29. What does CPCTC mean?
AD BC
AB DC and AD BC
30. Prove that
Given:
Statements
Reasons
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
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
4.7 Isosceles and Equilateral Triangles
32. What is the Base Angles Theorem?
33. Mark the base angles in the triangle.
34. If a triangle is equilateral then it must also be _____________________________.
In 35 – 39 solve for the missing variable.
35.
36.
37.
38.
SPIRAL REVIEW (YOU WILL BE QUIZZED OVER THIS MATERIAL)
1) Ray PR is shown in which sketch?
a)
b)
P
P
Q
R
Q
R
c)
d)
P
Q
R
R
Q
P
2) Use the Segment Addition Postulate to solve for p.
FE = 4p + 20
EG = 5p + 16
FG = 45
F
a) p = 1
b) p = 4
c) p = 5
E
G
d) p = 6
3) Find the distance between A( 6, 0) and B( 4, 4).
a) 116
b)
20
c)
116
d) 20
4) Which of the following angles could measure 125 ?
a)
b)
c)
d)
5) If m JOL = 50 and m KOL = 27 , then what is the measure of JOK?
a) 25
b) 23
c) 28
J
d) 20
K
O
L
6) In the figure below, m AED = 128 . Which of the following statements is false?
a) m BEC = 128
b) AEB and DEC are congruent
c) BEC and CED are vertical angles
d) m AEB = 52
A
D
E
B
7)
1 and 2 are ______.
a)
b)
c)
d)
C
8) Find the measure of 1.
1
a linear pair
complementary angles
supplementary angles
vertical angles
2
a)
b)
c)
d)
42
48
58
138
1
42
9) Solve for x.
a)
b)
c)
d)
5x + 24
3
27
75
39
10) Name an angle that is adjacent to
a)
b)
c)
d)
3x + 30
BOC
DOE
AOE
DOB
BOA
11) Using the diagram, name an angle that is complementary to
a)
AOE
b)
BOC
c)
DOE
d)
AOC
COD.
B
C
A
O
E
D
12) What is the converse of the statement, “If it rains then I carry my umbrella.”?
a)
b)
c)
d)
“If it does not rain, then I do not carry my umbrella.”
“If I do not carry my umbrella, then it does not rain.”
“If I do not carry my umbrella, then I will get wet.”
“If I carry my umbrella, then it rains.”
13) What is the inverse of the statement, “If two lines are parallel, then they do not intersect.”?
a)
b)
c)
d)
“If two lines are not parallel then they intersect.’
“If two lines intersect then they are not parallel.”
“If two lines do not intersect then they may be skew.”
“If two lines do not intersect then they are not parallel.”
14) Which of the following statements is false?
a)
b)
c)
d)
Three non-collinear points determine a plane.
Any three points are collinear.
A line contains at least two points.
Through any two distinct points there exists exactly one line.
15) In the cube shown at right, QT and YS are called _____
a)
b)
c)
d)
parallel lines
perpendicular lines
intersecting lines
skew lines
Use the diagram at right for questions 16 and 17.
l
16) In the figure shown, 1 and 2 are ____
a)
b)
c)
d)
17) If the m 1
80 , what must the m 3 be
In order for m
a)
b)
c)
d)
1
same-side interior angles
corresponding angles
alternate interior angles
alternate exterior angles
180
80
100
10
n ?
m
1
1
3
2
n