Exercise 54 Page 186

Let's add some labels to the given diagram.

Since ∠ ABC is marked as a right angle, we know that m∠ ABC=90. We can see that ∠ 3 and ∠ ABC form a linear pair. Therefore, the Supplement Theorem gives us the following relation. m∠ 3 + m∠ ABC = 180 We can find m∠ 3 by substituting m∠ ABC=90 into this equation.

m∠ 3 + m∠ ABC = 180

Substitute

m∠ ABC= 90

m∠ 3+ 90 = 180

SubEqn

LHS-90=RHS-90

m∠ 3= 90

Now, we can notice that ∠ 3, ∠ 4, and ∠ 5 are supplementary angles. Therefore, by the Supplement Theorem, we can write an equation. m∠ 3 + m∠ 4 + m∠ 5= 180 From the given statement we know that m∠ 4=32. Now, to find the measure of ∠ 5, we can substitute m∠ 3=90 and m∠ 4=32 into the equation above and solve it for m∠ 5.

m∠ 3 + m∠ 4 + m∠ 5= 180

SubstituteII

m∠ 3= 90, m∠ 4= 32

90 + 32 + m∠ 5= 180

AddTerms

Add terms

122 + m∠ 5=180

SubEqn

LHS-122=RHS-122

m∠ 5=58