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Chapter 4 – Congruent Triangles
4.2 and 4.9– Classifying Triangles and Isosceles, and Equilateral Triangles.
Match the letter of the figure to the correct vocabulary word in Exercises 1–4.
1. right triangle
__________
2. obtuse triangle
__________
3. acute triangle
__________
4. equiangular triangle
__________
Classify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two
classifications for Exercise 7.)
5.
6.
7.
For Exercises 8–10, fill in the blanks to complete each definition.
8. An isosceles triangle has ____________________ congruent sides.
9. An ____________________ triangle has three congruent sides.
10. A ____________________ triangle has no congruent sides.
Classify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two
classifications in Exercise 13.)
11.
12.
13.
1
Isosceles Triangles
Remember…
Isosceles triangles are triangles with two congruent sides.
The two congruent sides are called legs.
The third side is the base.
The two angles at the base are called base angles.
Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.
Converse is true!
2
Isosceles Triangles; Proving Triangles Congruent
Find the value of x.
1.
2.
50°
3.
5x
60°
21
x°
3x + 20
3x
4.
5.
100°
72°
x°
x°
3
4.3 Angle Relationships in Trinagles





o
The interior is the set of all points inside the figure. The
exterior is the set of all points
outside the figure.
An interior angle is formed by two sides of a triangle.
An exterior angle is formed by one side of the triangle
and extension of an adjacent side.
Each exterior angle has two remote interior angles. A
remote interior angle is an
interior angle that is not adjacent to the exterior angle.
Exterior Angles: Find each angle measure.
37. mB ___________________ 38.
mPRS
39. In LMN, the measure of an exterior angle at N measures 99.
1
mL  x 
2
3
and mM  x  . Find mL, mM, and mLNM.
3
40. mE and mG
__________________
41. mT and mV
____________________
___________________
42. In ABC and DEF, mA  mD and mB  mE. Find mF if an exterior
angle at A measures 107, mB  (5x  2) , and mC  (5x  5) .
_______________
43.
The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle.
____________________
44. One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
___________________
4
45. The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
46. The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
47.
Find mB
48. Find m<ACD
49. Find mK and mJ
50. Find m<P and m<T
Use the figure at the right for problems 1-3.
1. Find m3 if m5 = 130 and m4 = 70.
5
2. Find m1 if m5 = 142 and m4 = 65.
2
3
4
1
3. Find m2 if m3 = 125 and m4 = 23.
Use the figure at the right for problems 4-7.
11
4. m6 + m7 + m8 = _______.
5. If m6 = x, m7 = x – 20, and m11 = 80,
then x = _____.
6.
If m8 = 4x, m7 = 30, and m9 = 6x -20,
then x = _____.
9
8
6
7
10
7. m9 + m10 + m11 = _______.
For 8 – 12, solve for x.
x°
8.
x°
140°
9.
(5x)°
35°
120
5
4.4 Congruent Triangles
Congruent Triangles: Two ’s are
 if their can be matched up so
that corresponding angles and
sides of the ’s are .
Congruence Statement: \
RED  FOX
List the corresponding ’s:
corresponding sides:
R  ___
RE  ____
E  ___
ED  ____
D  ___
RD  ____
Examples:
1. The two ’s shown are .
a) ABO  _____
b) A  ____
c) AO  _____
d) BO = ____
C
D
O
K
2. The pentagons shown are .
a) B corresponds to ____
c) ______ = mE
A
B
L
The following ’s are , complete the congruence statement:
6
4 cm
H
R
e) If CA  LA , name two right ’s in the figures.
3. Given BIG  CAT, BI = 14, IG = 18, BG = 21, CT = 2x + 7. Find x.
E
C
B
b) BLACK  _______
d) KB = ____ cm
S
A
O
Use Graph paper to do the following:
7. Plot the given points. Draw BIG. Locate point P so that BIG  PIG.
a) B(1,2) I(4,7) G(4,2)
b) B(7,5)
I(-2,2) G(5,2)
Plot the given points on graph paper. Draw ABC and DEF. Copy and complete the statement
ABC  _____.
8. A(-1,2) B(4,2) C(2,4)
D(5,-1) E(7,1) F(10,-1)
9. A(-7,-3) B(-2,-3) C(-2,0)
D(0,1)
E (5,1) F(0,-2)
10. A(-3,1) B(2,1) C(2,3)
D(4,3) E(6,3) F(6,8)
11. A(1,1) B(8,1) C(4,3)
D(3,-7) E(5,-3) F(3,0)
Plot the given points on graph paper. Draw ABC and DE . Find two locations of point F
Such that ABC   DEF.
12. A(-1,0) B(-5,4) C(-6, 1)
D(1,0)
E(5,4)
Parts of a Triangle in terms of their relative positions.
1. Name the opposite side to C.
2. Name the included side between A and B.
A
3. Name the opposite angle to BC .
4. Name the included angle between AB and AC .
B
C
4.5-4.7 Proving Triangles Congruent
Ways to Prove ’s :
SSS Postulate: (side-side-side) Three sides of one  are  to three sides of a second ,
A
Given: AS bisects PW ; PA  AW
P
W
S
SAS Postulate: (side-angle-side) Two sides and the included angle of one  are  to two sides
and the included angle of another .
X
Given: PX bisects AXE; AX  XE
P
A
7
E
ASA Postulate: (angle-side-angle) Two angles and the included side of one  are  to two angles
and the included side of another .
Given:
MA //TH
A
T
AT // MH
M
H
AAS Theorem: (angle-angle-side) Two angles and a non-included side of one  are  to two
angles and a non-included side of another .
C
Given: UZ bi sects CA
UZ  CU ; UZ  ZA
R
U
Z
A
HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right  are  to the hypotenuse
and leg of another right .
A
Given: AT  FC
Isosceles  FAC with legs FA, AC
F
C
T
Congruent Triangles Examples
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must
correspond to your answer.
1.
A
G
E
D
I
C
3.
H
2.
B
F
Q
4.
R
10
A
B
12
12
T
C
S
R
5.
Q
S
D
V
6.
S
P
10
T
U
8
R
7.
8.
State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must
correspond to your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Fill in the congruence statement and then name the postulate that proves the ∆s are . If the ∆s are not , write “not possible” in
second blank. (Leave first blank empty)*Markings must go along with your answer**
Some may have more
E than one postulate
A
D
1.
2.
B
B
C
E
D
A
C
F
∆ABC  ________ by __________
∆ABC  _____ by ________
E
3.
F
A
4.
D
A
B
B
∆ABC  ________
by __________
F
C
∆ABC  ________ by _________D
9
C
5.
6.
D
B
B
P
Q
C
A
E
A
∆ABC  ________ by _________
C
R
∆ABC  _______ by __________
7.
C
8.
C
60
D
A
30
60
A
50
B
70
60
B
D
∆ABC  ________ by ____________
∆ABC  ________ by ___________
9.
10.
A
A
C
B
B
D
C
∆ABC  ________ by _________
11.
∆ABC  _______ by ___________
B
12.
C
A
D
A
D
B
C
D
∆ABC  _________ by ___________
A
∆ABC  _________ by __________
A
U
13.
14.
B
B
C
∆ABC  _________ by ___________
N
∆ABC  __________ by ___________
C
10
D
#1 Given: SR  UT ; SR // UT ; S  U
Prove: ST // UV
1. SR  UT ; SR // UT ; S  U
1. _____________________________
2. 1  4
2. __________________________________________
3. ∆RST  ∆TUV
3. __________________________________________
4. 3  2
4. __________________________________________
5. ST // UV
5. __________________________________________
#2 Given: D is the midpoint of AB; CA  CB
Prove: CD bisects ACB.
1. D is the midpoint of AB; CA  CB
1. _________________________________________
2. AD  DB
2. __________________________________________
3. CD  CD
3. __________________________________________
4. ∆ACD  ∆BCD
4. __________________________________________
5. 1  2
5. __________________________________________
6. CD bisects ACB.
6. __________________________________________
#3 Given: AR≅ AQ; RS ≅ QT
Prove: AS ≅ AT
1. AR≅ AQ; RS ≅ QT
1. ________________________
2. <R  <Q
2. __________________________________________
3. ARS  AQT
3. __________________________________________
4. AS ≅ AT
4. __________________________________________
11
D
Fill in Proofs:
#1
3 4
Given: AB  CB
AC  BD
Prove: Δ ADB  Δ CDB
1 2
A
C
B
1.
AB  CB
1. _________________________________________________
2.
AC  BD
2. _________________________________________________
3.  1 & 2 are right ’s.
3. _________________________________________________
4.  1  2
4. _________________________________________________
BD  BD
5. _________________________________________________
6. Δ ADB  Δ CDB
6. _________________________________________________
5.
#2
D
Given: AC  BD
BD bisects ADC
3 4
Prove: AB  CB
A
1.
AC  BD
1. _________________________________________________
2.  1 & 2 are right ’s
2. _________________________________________________
3. 1  2
3. _________________________________________________
4.
BD  BD
4. _________________________________________________
5.
BD bisects ADC
5. _________________________________________________
6.  3  4
6. _________________________________________________
7. Δ ADB  Δ CDB
7. _________________________________________________
AB  CB
8. _________________________________________________
8.
12
1 2
B
C
R
Congruent Triangles Proofs
1.
Given:
Prove:
S
P  S ; O is the midpoint of PS
O is the midpoint of RQ
O
Q
P
D
A
2.
Given:
Prove:
CD  AB ; D is the midpoint of AB
CA  CB
K
3.
B
Given:
SK // NR; SN // KR
Prove:
SK  NR; SN  KR
1
S
C
4
2
R
3
D
4.
Given:
Prove:
N
AD // ME; AD  ME
M is the midpoint AB
DM // EB
A
M
A
AE  BC
5.
Given:
AD  BD
Prove:
DE  DC
EDB  ADC
E
B
B
D
E
6.
Given:
AB  MK
B is the midpoint of
Prove:
C
A
x  y
MK
y
x
M
13
B
K
1
7.
Given:
CD  FM
1  2
F
D
C
Prove:
CD bisects MCF
M
2
S
8.
Given: PS  QS and PV  QV
Prove:
x  y
C
x
AC  BC
9.
Given:
Y
V
P
Q
1
CE  CD
AE  BD
Prove: 1  2
A
D
A
10.
Given:
E
D
Prove:
2
D  A
B
14
C
E
B