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Triangle Side Length and Angle Relationships

Explore the relationship between side lengths and angle measures in triangles. Identify the largest and smallest angles and sides, and understand how they relate to each other.

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Triangle Side Length and Angle Relationships

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  1. EXAMPLE 1 Relate side length and angle measure Draw an obtuse scalene triangle. Find the largest angle and longest side and mark them in red. Find the smallest angle and shortest side and mark them in blue. What do you notice? SOLUTION The longest side and largest angle are opposite each other. The shortest side and smallest angle are opposite each other.

  2. EXAMPLE 1 Relate side length and angle measure The relationships in Example 1 are true for all triangles as stated in the two theorems below. These relationships can help you to decide whether a particular arrangement of side lengths and angle measures in a triangle may be possible.

  3. Draw a diagram and label the side lengths. The peak angle is opposite the longest side so, by Theorem 5.10, the peak angle is the largest angle. EXAMPLE 2 Standardized Test Practice SOLUTION

  4. o The angle measures sum to 180 , so the third angle measure is 180 – (46 + 59 ) = 75 . You can now label the angle measures in your diagram. o o o o o The greatest angle measure is 75 , so the correct answer is C. EXAMPLE 2 Standardized Test Practice

  5. 1. List the sides of RSTin order from shortest to longest. SOLUTION ST, RS, TR for Examples 1 and 2 GUIDED PRACTICE

  6. Another stage prop is a right triangle with sides that are 6, 8, and 10feet long and angles of 90 , about 37 , and about 53 . Sketch and label a diagram with the shortest side on the bottom and the right angle at the left. o o o SOLUTION 2. for Examples 1 and 2 GUIDED PRACTICE

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