A triangle can be classified by _________ and __________. sidesangles There are four ways to classify triangles by angles. They are Equiangular Acute Obtuse Right All angles congruent 3 acute angles 1 obtuse and two acute angles 1 right and two acute angles There are three ways to classify triangles by sides. They are Equilateral - 3 congruent sides and angles Isosceles - 2 congruent sides and angles Scalene - no congruent sides or angles

Let’s take a closer look at the isosceles triangle: The two congruent sides are called legs. LEG The other side is called the base. BASE The two congruent angles are called the base angles. The other angle is called the vertex angle. VERTEX ANGLE BASE ANGLES

WHITE NOTE CARD: Isosceles Triangle LEG BASE VERTEX ANGLE BASE ANGLES The legs and base angles are congruent. Vertex angles is always opposite the base.

REVIEW: RIGHT TRIANGLE LEG HYPOTENUSE

Angle Sum Theorem What is the sum of the measures of the angles in a triangle? Is this true for all triangles? Even the really big ones and really small ones? Proof of Angle Sum Theorem

Remember the proof of the Triangle Sum Theorem? Good stuff! Given: ABC is a triangle. A B C (Number the angles for convenience) Prove:  1 +  2 +  3 = 180 STATEMENTS REASONS 1. ABC is a triangle 1. Given 2.Draw a line through B that is parallel to AC 2. Parallel Postulate 3.  4 +  2 +  5 =  1   4 and  3   5 5.  1 +  2 +  3 = If two parallel lines are cut by a transversal then the alternate interior angles are congruent. 5. Substitution Property Definition of Supplementary angles Click on the reason for an explanation.

An exterior angle is formed by one side of a triangle and the extension of another side. Exterior angle

Interior angles Remote interior angles are the interior angles in the triangle that are not adjacent to the exterior angle. Exterior angle remote interior angles Exterior angle

Exterior Angle Theorem The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles. Exterior angle Remote interior angles In other words,  4 =  1 +  2.

Add this proof to page 37 that already has the Triangle Sum Theorem proof. What could be more fun than another proof? I can’t wait!

Given: ABC is a triangle with exterior angle  4 A B C Prove:  1 +  2 =  4 (The sum of the measures of the two remote interior angles in a triangle is equal to the measure of the exterior angle) STATEMENTS REASONS 1.ABC is a triangle with exterior angle  4 2.  1 +  2 +  3 =  3 +  4 =  1 +  2 +  3 =  3 +  4 5.  1 +  2 =  4 1.Given 2.Triangle Sum Theorem 3.Definition of Linear Pair 4.Substitution Property 5.Subtraction Property  1 +  2 +  3 = 180

Add the Exterior Angle Theorem to your Triangle Sum Theorem note card (it’s a colored one). Triangle Sum Theorem The sum of the measures of the angles in a triangle is 180. Exterior Angle Theorem The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles  4 =  1 +  2

Proofs are awesome! I hope we get to do more of them soon! I’m sure we will! Geometry is super! I love math! It’s everywhere!