Two Proof-Oriented Triangle Theorems

1
Two Proof-Oriented Triangle Theorems
Lesson 7.2
2
Theorem 53 If 2 angles of one triangle are
congruent to two angles of a second triangle,
then the third angles are congruent. (no-choice
Theorem)
F
C
B
A
D
E
If ltA congruent ltD ltB congruent ltE Then ltC
congruent ltF Since the sum 180 subtract and get
ltC congruent ltF The triangles do not have to be
congruent, the angles do!
3
Theorem 54 If there exists a correspondence
between the vertices of two triangles such that
two angles and a non-included side of one
triangle are congruent to the corresponding parts
of the other, then the triangles are congruent.
(AAS)
4
J
Given JM ? GM GK ? KJ Conclude ltG ? ltJ
K
G
H
M
1. JM ? GM, GK ? KJ 2. ?GMJ, ?JKG rt ?s 3. ?GMJ
? ?JKG 4. ?GHM, ?JHK vert ?s 5. ?GHM ? ?JHK 6. ?G
? ?J

1. Given
2. ? lines from rt ?s
3. Rt ?s are ?
4. Assumed from diagram
5. Vert. ?s are ?
6. No Choice Theorem

5
Given Triangle as marked. Find the m ?1.
60
3x-5
x5
1
By Ext ? Theorem 3x 5 60 (x 5) 3x 5
65 x 2x 70 x 35
?1 is supp to (3x 5) Then ?1 (3x 5) 180
?1 3(35) 5 180 ?1 105 5 180 ?1 100
180 ?1 80