Exercise 23 Page 525

From the figure, we can identify a number of vertical angles. &∠ 1 and ∠ 5 &∠ 2 and 40^(∘) &∠ 3 and 30^(∘) According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Therefore, we know that ∠ 1≅ ∠ 5, m∠ 2=40, and m∠ 3=30^(∘). Let's show this in our diagram. Note that ∠ 1 is a right angle and therefore measures 90^(∘).

To find the remaining m∠ 4, notice that 30^(∘), 40^(∘), and ∠ 4 in the figure below make up a right angle.

By adding these three angle measures and setting their sum equal to 90^(∘), we can solve for m∠ 4.

30^(∘)+40^(∘)+m∠ 4=90^(∘)

AddTerms

Add terms

70^(∘)+m∠ 4=90^(∘)

AddTerms

Add terms

70^(∘)+m∠ 4=90^(∘)

SubEqn

LHS-70^(∘)=RHS-70^(∘)

m∠ 4=20^(∘)

The last angle has a measure of 20^(∘).