Congruent Triangles

Geometry Chapter 4

4.1 Triangles and Angles

Classification by Sides:

Triangles and Angles

Classification by Angles

Parts of Triangles

leg leg

base

leg

leg

hypotenuse

Vertex angle

Base angle Base angle

ExteriorangleInterior

angle

Theorems Involving Triangles

The sum of the measures of the angles of a triangle = 180°

The measure of the exterior angle of a triangle = the sum of the two remote interior angles. 12

3

A

B

C

Corollaries to Triangle Theorems

The acute angles of a right triangle are complementary.

Each angle of an equiangular triangle has a measure of 60°.

In a triangle, there can be at most one right angle or one obtuse angle.

¬

Examples

Sides of lengths 2mm, 3mm and 5mm.

Sides of lengths 3m, 3m, 3m.

Sides of lengths 8m, 8m, 5m.

Examples

Angles of measures 90, 25, 65.

Angles of measures 60, 60, 60.

Angles of measures 80, 70, 30.

Angles of measures 140, 30, 10.

Examples

A triangle has angles that measure x, 7x, and x. Find x.

Examples

A right triangle has angle measures of x and (2x-21). Find x.

Examples

Find the measure of the exterior angle shown.

4.2 Congruence and Triangles

Congruent – same size, same shape

Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent

A C

B

D

E

F

If ΔABC ΔDEF,thenA D AB DEB E BC EFC F AC DF

Theorems about Congruent Figures

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

If R M and S N, then T O

R

S

T M

N

O

Examples

110°

87°

10m

72°

L M

NO

If LMNO EFGH, find x and y.

E F

GH

(7y + 9)°

(2x +3)m

Examples

4.3-4.3 Proving Triangles Congruent

SSS – Side Side Side – If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

If AB DE BC EF AC DF, then

ABC DEF

SAS

SAS – Side Angle Side – If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

If AB DE BC EF B E, then ABC DEF

ASA

ASA – Angle Side Angle – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

If A D C F AC DF,then ABC DEF

AAS

AAS – Angle Angle Side – If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

If A D C F AB DE,then ABC DEF

HL

HL – Hypotenuse Leg – If the hypotenuse and leg of one RIGHT triangle are congruent to the hypotenuse and leg of another RIGHT triangle then the triangles are congruent.

C

A

B F

D

E

If ABC,DEF Right s,AB DE, AC DF, thenABC DEF.

4.5 Using Congruent Triangles

Definition of Congruent Triangles (rewritten)

Corresponding Parts of Congruent

Triangles are CongruentCPCTC is used often in proofs involving

congruent triangles.

1.

M

TS

R

AA is the midpoint of MT.A is the midpoint of SR.

MS ll TR

A is the midpoint of MT.A is the midpoint of SR.

1. Given

UR ll ST R and T are right angles

UR ll ST R and T are right angles

1. 1. Given

U

R S

T

4.6 Isosceles, Equilateral and Right Triangles

If two sides of a triangle are congruent, then the angles opposite are congruent. (Base angles of an isosceles triangle are congruent.

Converse – If two angles of a triangle are congruent, then the sides opposite are congruent.

A

B

C

If BA BC, then A C.

If A C, then BA BC.

More Corollaries

If a triangle is equilateral, then it is equiangular.

If a triangle is equiangular then it is equilateral.

A

B

C

Examples

Find x and y.

35 x

y

Examples

Find the unknown measures.

50 ?

?

Examples

Find x.

33 in(x-11) in

Examples

Find x and y.

y

40

x