Exercise 15 Page 616

Let's illustrate our right triangle in a diagram.

Examining our diagram, ∠ C, is going to be the angle that's 4 times the difference of the other acute angle and 5. Thus, the phrase "the measure of the other acute angle" refers to the smaller of the non-right angles, ∠ B. Now we know enough to translate the verbal expression into a mathematical one: 4 times the difference of the measure of the other acute angle and 5 4( m∠ B - 5^(∘)) Hence, this expression equals the measure of ∠ C. According to the Corollary to the Triangle Sum Theorem, the acute angles of a right triangle are complementary. This means the non-right angles add up to 90^(∘). m∠ B+m∠ C=90^(∘) Since we have an expression for m∠ C in terms of m∠ B, we can substitute m∠ C with this expression and solve for m∠ B.

m∠ B+m∠ C=90^(∘)

Substitute

m∠ C= 4(m∠ B-5^(∘))

m∠ B+ 4(m∠ B-5^(∘))=90^(∘)

▼

Solve for m∠ B

Distr

Distribute 4

m∠ B+4m∠ B-20^(∘)=90^(∘)

AddTerms

Add terms

5m∠ B-20^(∘)=90^(∘)

AddEqn

LHS+20^(∘)=RHS+20^(∘)

5m∠ B=110^(∘)

DivEqn

.LHS /5.=.RHS /5.

m∠ B=22^(∘)

If the smaller of our acute angles is 22^(∘) the larger one must be 90^(∘)-22^(∘)=68^(∘).