Exercise 29 Page 340

There is no direct connection between the given angles and angle ∠ 6, so we will find m∠ 6 in two steps. Let's find m∠ 5 first.

Finding m∠ 5

Paying attention the the denoted angle marks, note that ∠ 5 is one of the acute angles of a right triangle. What other information is given? The other acute angle! We can use both given measures to find the third measure of a triangle.

According to Corollary 4.1 of the Triangle Angle-Sum Theorem, the acute angles of a right triangle are complementary, so their measures add to 90^(∘). The measure of ∠ 5 is 55. Great find.

Finding m∠ 6

Let's put the measure of angle ∠ 5 we just found on the diagram. Notice that angles ∠ 5 and ∠ 6 together form a right angle, so these are complementary angles.

The sum of the measures of complementary angles is 90^(∘). Since we know the measure of ∠ 5, this lets us find the measure of ∠ 6. The measure of ∠ 6 is 35^(∘).