4.1.1 Classifying Triangles (continued)

The figure given here is a large triangle, △ ABD , that is divided into two smaller triangles, △ ACD and △ ACB , by AC . Notice that the figure contains a right angle mark at ∠ DAB . Therefore, m ∠ DAB = 90 ° . The first triangle to classify is △ ACB . Since two of its angles are given to be 60 ° , the measure of only one angle, m ∠ CAB , must be found in order to classify △ ACB by its angles. By the Angle Addition Postulate, m ∠ DAC + m ∠ CAB = m ∠ DAB , or m ∠ CAB = m ∠ DAB − m ∠ DAC . It is given in the figure that m ∠ DAB = 90 ° and m ∠ DAC = 30 ° . So, m ∠ CAB = 90 ° − 30 ° = 60 ° . Therefore, the measure of each angle in △ ACB is 60 ° , and so all three angles are congruent. Thus, this is an equiangular triangle. Note that because the measures of all three angles in △ ACB are less than 90 ° , △ ACB is also an acute triangle. Classify △ ACD . Two of its angle measures are known: m ∠ ADC = 30 ° and m ∠ DAC = 30 ° . So, find m ∠ ACD . Since ∠ ACD and ∠ ACB form a straight angle, m ∠ ACD + m ∠ ACB = 180 ° . Therefore, m ∠ ACD = 180 ° − 60 ° = 120 ° , which is an obtuse angle. So, this is an obtuse triangle.

Example 2 Classifying Triangles by Side Lengths A triangle can be classified by its side lengths. When a triangle is classified by its side lengths, there are three possibilities: scalene (no congruent sides), isosceles (at least two congruent sides), and equilateral (three congruent sides).

There are two triangles in the figure given here, △ SPR and △ TPQ . Each triangle will be classified by its side lengths. First, classify △ TPQ . Notice that the triangle contains single tick marks on PT , PQ , and TQ . These tick marks indicate that those three sides are congruent. So, all three sides of △ TPQ are congruent. A triangle with three congruent sides is classified as equilateral. Therefore, △ TPQ is an equilateral triangle.

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