Exercise 1 Page 362

We are asked to explain how angle measures can be used to determine whether two triangles are similar. Imagine we are given two triangles and know some of their angle measures. In this situation, we can use the Angle-Angle (AA) Criterion.

This lets us confirm that two triangles are similar by finding two pairs of corresponding angles that are congruent. Let's consider two example triangles.

Right away we can see one pair of congruent angles of measure 82^(∘).

The other two given angles are not congruent. However, we still do not know the measures of two angles. Let's call one of the missing measures x.

We know that the sum of the measures of internal angles of a triangle is 180^(∘). This lets us write an equation. 82^(∘) + 60^(∘) + x = 180^(∘) Let's solve this equation for x!

82^(∘) + 60^(∘) + x = 180^(∘)

AddTerms

Add terms

142^(∘) + x = 180^(∘)

SubEqn

LHS-142^(∘)=RHS-142^(∘)

142^(∘) + x - 142^(∘) = 180^(∘) - 142^(∘)

SubTerms

Subtract terms

x = 38^(∘)

The missing measure is 38^(∘).

As we can see, there are now two corresponding pairs of congruent angles. By the Angle-Angle Criterion, the two triangles are similar.