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Warm-Up Exercises 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 ~ 2? 1= 2 ANSWER Alt. Int. s Thm. 1 Warm-Up Exercises Target Classify Triangles by Sides and Angles You will… Classify triangles and find measures of their angles. Warm-Up Exercises 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 Warm-Up Exercises Vocabulary • interior angles – original angles of the triangle – “inside” • exterior angles – the angles that form linear pairs with the interior angles of a triangle Warm-Up Exercises 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 Warm-Up Exercises 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 Warm-Up2Exercises EXAMPLE 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 = = ( x2 – x1 ) 2 + ( y2 – y1 ) 2 ( (– 1 ) – 0 ) 2 + ( 2 – 0 ) 2 = 5 2.2 = 45 6.7 ( 6 – (– 1 )) 2 + ( 3 – 2 ) 2 = 50 7.1 OQ = ( x2 – x1 ) 2 + ( y2 – y1 ) 2 = ( 6 – 0 )2 + ( 3 – 0 )2 PQ = ( x2 – x1 ) 2 + ( y2 – y1 ) 2 = Warm-Up2Exercises EXAMPLE Classify a triangle in a coordinate plane STEP 2 Check for right angles. 2–0 The slope of OP is = – 2. – 1– 0 The slope of OQ is 3 – 0 = 1 . 2 6–0 1 The product of the slopes is – 2 = – 1, 2 so OP OQ and ANSWER Therefore, POQ is a right angle. PQO is a right scalene triangle. Warm-Up Exercises GUIDED PRACTICE 1. A for Examples 1 and 2 Draw an obtuse isosceles triangle and an acute scalene triangle. B C obtuse isosceles triangle Q R P acute scalene triangle Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 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. BC = ( ( 3 ) – 0 ) 2 + ( 3 – 0 ) 2 = 18 ( ( 3 ) – 3 ) 2 + ( −3 – 3 ) 2 = 36 AC = (( 0 ) – 3 )2 + ( 0 – 3 )2 = AB = 3–0 = 1. 3–0 3–3 The slope of BC is = 0. −3 – 3 3–0 The slope of AC is = −1. −3 – 0 The slope of AB is ANSWER ABC is isosceles 18 BAC is a right angle ABC is a right Isosceles triangle. Warm-Up3Exercises EXAMPLE 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° x = 75 Apply the Exterior Angle Theorem. Solve for x. STEP 2 Substitute 75 for x in 2x – 5 to find m JKM. 2x – 5 = 2 75 – 5 = 145 ANSWER The measure of JKM is 145°. Warm-Up4Exercises EXAMPLE Find angle measures from a verbal description Use the corollary to set up and solve an equation. x° + 2x° = 90° x = 30 Corollary to the Triangle Sum Theorem Solve for x. ANSWER So, the measures of the acute angles are 30° and 2(30°) = 60° . Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 3. Find the measure of 1 in the diagram shown. ANSWER The measure of 1 in the diagram is 65°. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 4. Find the measure of each interior angle of ABC, where m A = x ° , m B = 2x° , and m C = 3x°. SOLUTION m A+m B+m C= 180° Apply Triangle Sum Theorem x + 2x + 3x = 180° 6x = 180° x = 30° B : 2x = 2(30) = 60° C : 3x = 3(30) = 90° Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 5. Find the measures of the acute angles of the right triangle in the diagram shown. ANSWER 26° and 64° 6. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? ANSWER ACD =150°. A 2x B x C Q