Let x be the smaller of the measures of the angles. Use the given ratio to rewrite the measures of the other two angles in terms of x. Did you get a special right triangle? If so, use the corresponding theorem.
37.9
Practice makes perfect
Let x, y, and z be the measures of the angles of the triangle. By the Triangle Angle-Sum Theorem, we have the following equation.
x + y + z = 180^(∘)
We are told that the ratio of the measure of the angles is 1:2:3, which leads us to the following relations.
x = x y = 2x z = 3xLet's substitute the previous expressions into the first equation.
x + 2x + 3x = 180^(∘) ⇒ x = 30^(∘)
From the above, we conclude that the measure of the angles of our triangle are x=30^(∘), y=60^(∘), and z=90^(∘). Therefore, our triangle is a 30^(∘)-60^(∘)-90^(∘) triangle.
We also know that the shortest side is 8 units long. Therefore, the side opposite the 30^(∘) angle is 8 units long.
