Chapter 4: Triangle Congruence

Lindsey Sholander and Anna Schwarz

4-1: Triangle Classifications

By Angle Measures:• An Acute Triangle has three acute angles• An Obtuse Triangle has one obtuse angle• A Right Triangle has one right angle

4-1: Triangle Classifications

By Side Lengths:• An Isosceles Triangle has at least two

congruent sides• A Scalene Triangle has no congruent sides• An Equilateral Triangle has three congruent

sides

4-1: Triangle Classifications

Remember: When you look at a figure you cannot assume segments are congruent based on their appearance. They must be marked as congruent.

4-2: Angle Relationships in Triangles

• Triangle Sum Theorem: The sums of the angle measures of a triangle is 180⁰

• An auxiliary line is a line added to a figure to aid in a proof

4-2: Angle Relationships in Triangles

Corollaries:• The acute angles of a right triangle are

complementary• The measure of each angle of an equiangular

triangle is 60⁰

4-2: Angle Relationships in Triangles

Exterior Angle Theorem:The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.A remote interior angle is an interior angle that is not adjacent to the exterior angle

4-3: Congruent Triangles

Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent.

4-4: Triangle Congruence: SSS and SAS

Side, Side, Side (SSS) Postulate:If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

4-4: Triangle Congruence: SSS and SAS

Side, Angle, Side (SAS) Postulate:If two sides and the included angle of one triangle are congruent to two sides of another triangle then the triangles are congruent

4-5: Triangle Congruence: ASA, AAS and HL

Angle, Side, Angle (ASA) Postulate:If two angles and the included side of one triangle are congruent to two angles and the included side of an another triangle, then the triangles are congruent.

4-5: Triangle Congruence: ASA, AAS and HL

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

4-5: Triangle Congruence: ASA, AAS and HL

Hypotenuse-Leg (HL) TheoremIf the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

4-6: Triangle Congruence: CPCTC

CPCTC is an abbreviation for the phrase “Corresponding parts of congruent triangles are congruent.”

4-7: Introduction to Coordinate Proof

A coordinate proof is a style of proof that uses coordinate geometry and algebra.

4-7: Introduction to Coordinate Proof

Strategies for positioning figures in the coordinate plane:• Use the origin as a vertex, keeping the figure

in Quadrant I• Center the figure at the origin• Center a side of the figure at the origin• Use one or both axis’ as sides of the figure

4-8: Isosceles and Equilateral Triangles

The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base. The base angles are two angles that have the base as a side.

4-8: Isosceles and Equilateral Triangles

Isosceles Triangle Theorem:If two sides of a triangle are congruent then the angles opposite the sides are congruent.

4-8: Isosceles and Equilateral Triangles

Converse of Isosceles Triangle Theorem:If two angles of a triangle are congruent, the sides opposite those angles are congruent.

4-8: Isosceles and Equilateral Triangles

• Equilateral Triangle Corollary:If a triangle is equilateral, then it is equiangular• Equiangular Triangle Corollary:If a triangle is equiangular, then it is equilateral