4.1 Apply Triangle Sum Properties Hubarth Geometry

Classification of Triangles By Sides Equilateral Triangle Isosceles Triangle Scalene Triangle 3 congruent sides 2 congruent sides No congruent sides

Classification of Triangles By Angles Equiangular Triangle Acute Triangle 3 congruent angles 3 acute angles Right Triangle Obtuse Triangle 1 right angle 1 obtuse angle

Ex 1 Classify Triangles by Sides and By Angles Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55° , 55° , and 70° . It is an acute isosceles triangle. 55 55

Ex 2 Classify Triangles in a Coordinate Plane Classify PQO by its sides. Then determine if the triangle is a right triangle. STEP 1 Use the distance formula to find the side lengths. OP = y 2 – 1 ( ) x + = 2 – ( ) (– 1 ) + 5 2.2 OQ = y 2 – 1 ( ) x + 2 = – ( ) 6 + 3 45 6.7 PQ = y 2 – 1 ( ) x + 3 – 2 ( ) 6 + = (– 1 ) 50 7.1

Ex 2 Continued STEP 2 Check for right angles. The slope of OP is 2 – 0 – 1 – 0 = – 2. The slope of OQ is 3 – 0 6 – 0 = 2 1 . 1 The product of the slopes is – 2 2 = – 1 , so OP OQ and POQ is a right angle. Therefore, PQO is a right scalene triangle.

Angles When the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles. Exterior Angles Interior Angles Theorem Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180. B A C

Theorem Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent (remote- interior) interior angles. B 1 A C Ex 3 Find Angle Measures 2x – 5 = 70 + x x = 75 2x – 5 = 2(75) - 5 = 150-5 = 145

A Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. B C Ex 4 Find angle Measures From a Verbal Description The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle. xx 2x x + 2x = 90 3x = 90 x x=30 The two acute angles are 30 and 60

1. Draw an obtuse isosceles triangle and an acute scalene triangle. Practice 1. Draw an obtuse isosceles triangle and an acute scalene triangle. acute scalene triangle P Q R obtuse isosceles triangle B A C 2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle. ABC is a right Isosceles triangle. 3. Find the measure of 1 in the diagram shown. The measure of ∠ 1 in the diagram is 65°. 4. Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°. A x C 3x 2x B 5. Find the measures of the acute angles of the right triangle in the diagram shown. 26° and 64°