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Classifying Triangles by Sides and Angles: A Comprehensive Guide

This guide illustrates how to classify triangles based on their sides and angles using both geometric properties and coordinate plane methods. It includes examples demonstrating how to identify isosceles, scalene, acute, obtuse, and right triangles. The guide walks you through the process of applying the distance formula to find side lengths, measuring angles, and checking slopes to determine if a triangle is a right triangle. Whether for educational purposes or personal study, this resource simplifies the classification of triangle shapes.

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Classifying Triangles by Sides and Angles: A Comprehensive Guide

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

  2. Classify PQOby its sides. Then determine if the triangle is a right triangle. Use the distance formula to find the side lengths. STEP1 2 2 – – ( ( ) ) OP = + + 2 2 – – ( ( ) ) y x x y y y x x 2 1 2 2 2 1 1 1 2 2 ( – ( ) (– 1 ) ) 0 2 – 0 2.2 + = = 5 OQ = 2 2 ( – ( ) 6 ) 0 – 0 3 6.7 + = = 45 EXAMPLE 2 Classify a triangle in a coordinate plane SOLUTION

  3. PQ = 2 2 ( – ) 6 (– 1 ) ) 3 – ( 2 7.1 + = = Check for right angles. STEP2 The slope ofOPis 2 – 0 3 – 0 1 . – 2. The slope ofOQis = = – 2 – 0 2 6 – 0 1 – 2 The product of the slopes is – 1 , = 2 – ( ) 2 + 2 – ( ) so OPOQand POQ is a right angle. y x x y 2 2 1 1 50 ANSWER Therefore, PQOis a right scalene triangle. EXAMPLE 2 Classify a triangle in a coordinate plane

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

  5. Triangle ABChas 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. ANSWER ABCis a right Isosceles triangle. for Examples 1 and 2 GUIDED PRACTICE

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