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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.
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Isosceles Triangles
Remember…
Isosceles triangles are triangles with at least 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!
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4
Find the value of x.
1. 2. 3.
4. 5.
x°
50° 5x
3x + 20
21
3x
60°
100°
x°
72°
x°
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Corollary 4.3 Angle Relationships in Triangles
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. It forms alinear pair
with an angle of the triangle.
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. mB ___________________ 38. mPRS
39.In LMN, the measure of an exterior angle at N measures 99.
1m
3L x
and 2
m3
M x . Find mL, mM, and mLNM. ____________________
40.mE and mG ___________________ 41. mT and mV ___________________
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42. In ABC and DEF, mA mD and mB mE. Find mF if an exterior
angle at A measures 107, mB (5x 2) , and mC (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?
___________________
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 mB 48. Find m<ACD
49. Find mK and mJ 50. Find m<P and m<T
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Use the figure at the right for problems 1-3.
1. Find m3 if m5 = 130 and m4 = 70.
2. Find m1 if m5 = 142 and m4 = 65.
3. Find m2 if m3 = 125 and m4 = 23.
Use the figure at the right for problems 4-7.
4. m6 + m7 + m8 = _______.
5. If m6 = x, m7 = x – 20, and m11 = 80,
then x = _____.
6. If m8 = 4x, m7 = 30, and m9 = 6x -20,
then x = _____.
7. m9 + m10 + m11 = _______.
For 8 – 12, solve for x.
8.
9.
1 3
5 2 4
6
8
7 10
11
9
120
x°
(5x)°
x° 140°
35°
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4.4 Congruent Triangles
Polygons are congruent if all
of their corresponding sides
and all of their corresponding
angles are congruent.
Consecutive vertices of a polygon–
the endpoints of a side
Ex. P and Q are consecutive
vertices
Opposite vertices of a polygon-
vertices that are not consecutive
Congruent riangles: Two ’s are if they can be matched up so that corresponding angles and sides of the ’s are .
Congruence Statement: A congruence statement matches up the parts in the same order.
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 = ____
2. The pentagons shown are .
a) B corresponds to ____ b) BLACK _______
c) ______ = mE d) KB = ____ cm
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.
D C
O
B A
B
L A
C
K
H
O
R
S
4 cm
E
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The following ’s are , complete the congruence statement:
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.
3. Name the opposite angle to BC .
4. Name the included angle between AB and AC .
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 ,
Given: AS bisects PW ; AWPA
SAS Postulate: (side-angle-side) Two sides and the included angle of one are to two sides
and the included angle of another .
Given: PX bisects AXE; XEAX
ASA Postulate: (angle-side-angle) Two angles and the included side of one are to two angles
and the included side of another .
Given: MHAT
THMA
//
//
A
B
C
A
P W S
A
X
P
E
A
M
T
H
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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 .
Given: CAtsbiUZ sec
ZAUZCUUZ ;
HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right are to the hypotenuse
and leg of another right .
Given: FCAT
Isosceles FAC with legs ACFA,
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. 2.
3. 4.
5. 6.
7. 8.
C
R U Z
A
A
F T C
A
B
D
C
E G
H
F
I
T
Q
S
R
10
A B
D C
10
12 12
R
P Q
S
S T
V
R
U
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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 than one postulate
1. 2.
∆ABC _____ by ________ ∆ABC ________ by __________
3. 4.
∆ABC ________ by __________ ∆ABC ________ by _________
D
E
F
A
B C
A
B
C
D
F E
A
B C
D
E
F
A
B C
D
12
5. 6.
∆ABC ________ by _________ ∆ABC _______ by __________
7. 8.
∆ABC ________ by ___________ ∆ABC ________ by ____________
9. 10.
∆ABC ________ by _________ ∆ABC _______ by ___________
11. 12.
∆ABC _________ by ___________ ∆ABC _________ by __________
13. 14.
∆ABC _________ by ___________ ∆ABC __________ by ___________
A
B
C
D
E A
B
C
P Q
R
60
70
50
60
C D
A B
30
60 A
B
C
D
A B
C D
A
B C
D
C
A
B
D
B
A
C
U
N
A
B
C
D
A
D
B
C
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Proofs!
#1 Given: USUTSRUTSR ;//; Prove: UVST //
1. USUTSRUTSR ;//; 1. _____________________________
2. 1 4 2. __________________________________________
3. ∆RST ∆TUV 3. __________________________________________
4. 3 2 4. __________________________________________
5. UVST // 5. __________________________________________
#2 Given: D is the midpoint of CBCAAB ; Prove: CD bisects ACB.
1. D is the midpoint of CBCAAB ; 1. _________________________________________
2. DBAD 2. __________________________________________
3. CDCD 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. __________________________________________
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#1
Given: AB CB
AC BD
Prove: Δ ADB Δ CDB
1. AB CB 1. _________________________________________________
2. AC BD 2. _________________________________________________
3. 1 & 2 are right ’s. 3. _________________________________________________
4. 1 2 4. _________________________________________________
5. BD BD 5. _________________________________________________
6. Δ ADB Δ CDB 6. _________________________________________________
#2
Given: AC BD
BD bisects ADC
Prove: AB CB
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. _________________________________________________
8. AB CB 8. _________________________________________________
B
3 4
1 2
D
A C
D
B
3 4
1 2 A C
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Congruent Triangles Proofs
1. Given: SP ; O is the midpoint of PS
Prove: O is the midpoint of RQ
2. Given: ABCD ; D is the midpoint of AB
Prove: CBCA
3. Given: KRSNNRSK //;//
Prove: KRSNNRSK ;
4. Given: MEADMEAD ;//
M is the midpoint AB
Prove: EBDM //
5. Given: MKAB
B is the midpoint of MK
Prove: yx
6. Given: 21
FMCD
Prove: CD bisects MCF
A B
C
D
S
K
N
R
1
2 3
4
A M B
E D
A
B M K
x y
C D
M
F
1
2
P
O
R S
Q