Post on 29-Dec-2015
Triangle Congruencies
Lesson 4.4
c) What is PZ?
d) What is <Z
e) What is < N
f) What percent of M&Ms are brown?
Opener
a) Can you make a triangle using the sides: 5 ft., 12 ft., and 4 ft.?
b) One side of a triangle is 12 mm. Another side is 17 mm. What are the possible lengths for the third side?
Given PQZ
CNB
Z
PQ
C
N
B
10
333°
62°
Isosceles Triangle Conjecture
If a triangle is isosceles, then base angles are equal.
80° 80°
20°
Isosceles Triangles Conjecture
Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.
20°
25°45°
Exterior Angle Conjecture
Classwork Review
GEA
NCA
IECJAN
Classwork Review
Lesson 4.4 - Congruent TrianglesIs this a guarantee of triangle congruence?One pair of congruent sides
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?One pair of congruent sides
NO!
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides
NO!
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Three pairs of congruent sides
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Three pairs of congruent sides
YES!
SSS
90°65°25°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Three pairs of congruent angles
90°
65°
25°25°90°
65°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Three pairs of congruent angles
NO!
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesAre these guarantees of triangle congruence?
25°
SSSAAASASSSAASASAA
√
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?One pair of congruent angles, two pairs of congruent sides
25°
25°√
SSSAAASASSSAASASAA
√
25°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and the included angle
YES!SAS!
A
BC
ABC DEF
92°
D
E
F
92°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and the included angle
√
SSSAAASASSSAASASAA
√
An included angle is between the two given sides.
65°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and a non-included angle
√
SSSAAASASSSAASASAA
√
65°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and a non-included angle
√
SSSAAASASSSAASASAA
√
65°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and a non-included angle
√
SSSAAASASSSAASASAA
√
65°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and a non-included angle
√
SSSAAASASSSAASASAA
√
A
BC
D
E
F
ABC DEF
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent sides and a non-included angle
√
SSSAAASASSSAASASAA
√
NO!
90°
25°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent angles and the included side
√
SSSAAASASSSAASASAA
√
90°25°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent angles and the included side
√
SSSAAASASSSAASASAA
√
YES! ASA!
65°55°
A
B
C
65°55°
D
E
F
ABC DEF
ASA!
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent angles and the included side
√
SSSAAASASSSAASASAA
√
√
90° 35°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent angles and the non-included side
√
SSSAAASASSSAASASAA
√
√
35°
90°
35°
35°
Notes - Congruent TrianglesIs this a guarantee of triangle congruence?Two pairs of congruent angles and the non-included side
√
SSSAAASASSSAASASAA
√
√√
√
SSSAAASASSSAASASAA
√
√√
Notes - Congruent TrianglesThese are the triangle congruencies.
SSS
SAS
ASASAA
Using Properties of Right triangles
Theorem 4.8 Hypotenuse –Leg Congruence Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.
Using Properties of Right triangles
HL (Hypotenuse - Leg) is not like any of the previous congruence postulates... actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate.
• HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!!
SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
HL If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.
Methods of Proving Triangles Congruent
Congruent TrianglesName the congruence
ASA
FRS PQD
S
F
R
120°
35°D
P
Q120°
35°
SSA
FRS QSR
S
F
R
42°
42°Q
Congruent TrianglesName the congruence
SAS
FRS QSR
S
F
R
50°
50°Q
Congruent TrianglesName the congruence
SSS
MNR PTB
M
N
R
T
P
B
Names of the triangles in the congruence statement are not in corresponding order.
WHY?
Congruent TrianglesName the congruence
SSS
ASA