Classify each angle as acute, obtuse, or right. 1. 90º ANSWER right 2. 72º ANSWER acute 3. 116º ANSWER obtuse 4. How do you know that 1 = 2? ~ 2 1 ANSWER Alt. Int. s Thm.

Classify Triangles by Sides and Angles Target Classify Triangles by Sides and Angles You will… Classify triangles and find measures of their angles.

Vocabulary triangle – a polygon with 3 sides To classify by sides: scalene – no congruent sides isosceles – at least 2 congruent sides equilateral – exactly 3 congruent sides To classify by angles: acute – exactly 3 acute angles right – exactly 1 right angle obtuse – exactly 1 obtuse angle equiangular – exactly 3 congruent angles

Vocabulary interior angles – original angles of the triangle – “inside” exterior angles – the angles that form linear pairs with the interior angles of a triangle

Vocabulary Triangle Sum Theorem 4.1 – the sum of the measures of the interior angles of any triangle is 180° Corollary – the acute angles of a right triangle are complementary

Vocabulary Exterior Angle Theorem 4.2 – the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles m A + m B = m 1

EXAMPLE 1 Classify triangles by sides and by angles Support Beams Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. SOLUTION 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.

Classify a triangle in a coordinate plane EXAMPLE 2 Classify a triangle in a coordinate plane Classify PQO by its sides. Then determine if the triangle is a right triangle. SOLUTION 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

EXAMPLE 2 Classify a triangle in a coordinate plane 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. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Draw an obtuse isosceles triangle and an acute scalene triangle. obtuse isosceles triangle B A C acute scalene triangle P Q R

ABC is a right Isosceles triangle. GUIDED PRACTICE for Examples 1 and 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 right. = 3 – ( ) 2 ( 3 ) + 18 AB ABC is isosceles = −3 – 3 ( ) 2 ( 3 ) + 36 BC = – 3 ( ) 2 ( 0 ) + 18 AC The slope of AB is 3 – 0 = 1. BAC is a right angle The slope of BC is 3 – 3 −3 – 3 = 0. The slope of AC is 3 – 0 −3 – 0 = −1. ANSWER ABC is a right Isosceles triangle.

Write and solve an equation to find the value of x. EXAMPLE 3 Find an angle measure Find m∠ JKM. SOLUTION STEP 1 Write and solve an equation to find the value of x. (2x – 5)° = 70° + x° Apply the Exterior Angle Theorem. x = 75 Solve for x. STEP 2 Substitute 75 for x in 2x – 5 to find m∠ JKM. 2x – 5 = 2 75 – 5 = 145 The measure of ∠ JKM is 145°. ANSWER

EXAMPLE 4 Find angle measures from a verbal description ARCHITECTURE 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. SOLUTION First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

Find angle measures from a verbal description EXAMPLE 4 Find angle measures from a verbal description Use the corollary to set up and solve an equation. x° + 2x° = 90° Corollary to the Triangle Sum Theorem x = 30 Solve for x. So, the measures of the acute angles are 30° and 2(30°) = 60° . ANSWER

GUIDED PRACTICE for Examples 3 and 4 Find the measure of 1 in the diagram shown. The measure of ∠ 1 in the diagram is 65°. ANSWER

GUIDED PRACTICE for Examples 3 and 4 Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°. ° SOLUTION Apply Triangle Sum Theorem m A + m B + m C = 180° x + 2x + 3x = 180° 6x = 180° x = 30° B : 2x = 2(30) = 60° C : 3x = 3(30) = 90°

GUIDED PRACTICE for Examples 3 and 4 Find the measures of the acute angles of the right triangle in the diagram shown. 26° and 64° ANSWER In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? A B C Q 2x x ACD =150°. ANSWER