Congruent Triangles

Congruent Triangles Geometry Chapter 4

4.1 Triangles and Angles • Classification by Sides:

Triangles and Angles • Classification by Angles

Parts of Triangles Exterior angle Interior angle Vertex angle leg hypotenuse leg leg Base angle Base angle leg base

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. B C A 3 1 2

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 E B • Congruent – same size, same shape • Congruent Polygons(Triangles) – Two polygons (triangles) are congruent iff their corresponding sides and corresponding angles are congruent A C F D If ΔABC ΔDEF, then A D AB  DE B E BC  EF C 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 S N R M O T

Examples H G (2x +3)m L M 110° (7y + 9)° F 87° 72° E N O 10m If LMNO  EFGH, find x and y.

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 D A • 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. B E C F If ABC,DEF Right s, AB  DE, AC  DF, then ABC  DEF.

4.5 Using Congruent Triangles • Definition of Congruent Triangles (rewritten) Corresponding Parts of Congruent Triangles are Congruent CPCTC is used often in proofs involving congruent triangles.

M R A is the midpoint of MT. A is the midpoint of SR. A S MS ll TR T 1. A is the midpoint of MT. A is the midpoint of SR. 1. Given

U T UR ll ST R and T are right angles R S 1. UR ll ST R and T are right angles 1. Given

4.6 Isosceles, Equilateral and Right Triangles B • 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. C A 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. B A C

Examples • Find x and y. y 35 x

Examples • Find the unknown measures. ? 50 ?

Examples • Find x. (x-11) in 33 in

Examples • Find x and y. y 40 x

Congruent Triangles