Unit 3, Lesson 1
Angles and Steepness
Integrated Math 2
1.1
Warm-up
Triangle is similar to triangle . Write as many equations as you can to describe the relationships between the sides and angles of the two triangles.
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Triangle is similar to triangle . Write as many equations as you can to describe the relationships between the sides and angles of the two triangles.
Find the values of and . If there is not enough information, what else do you need to know?
Because of the Pythagorean Theorem, if we know any two sides of a right triangle, we can calculate the length of the third side. But what if we know a side and an angle rather than two sides?
All right triangles with one pair of congruent acute angles are similar by the Angle-Angle Triangle Similarity Theorem. Knowing just one side length in addition to those angle measures is enough to uniquely define the triangle.
The Americans with Disabilities Act includes guidelines for safe and accessible wheelchair ramps. Ramps must form a maximum 4.8-degree angle with the ground, which creates a maximum ratio for the legs of the right triangle.
Let's assume we are building a ramp for a 3-inch threshold.
To find length , we can use similarity. By corresponding sides, , so is 36 units. To find length , we can use the Pythagorean Theorem: . So , or about 36.1 units. To build a ramp that goes up 3 inches, we need to start 36 inches, or 3 feet, out from the edge of the threshold and use a board that's about 36.1 inches long.