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