Exercise 10 Page 111

Looking at the figure, we notice three things that we need to figure out the remaining angles.

1. ∠ 4 and ∠ 2 are vertical angles.
2. ∠ 4 is supplementary to both ∠ 1 and ∠ 3.
3. ∠ 1 and ∠ 3 are vertical angles.

By the Vertical Angles Congruence Theorem, ∠ 4 and ∠ 2 are congruent. m∠ 4 =m∠ 2 As m∠ 4 =29^(∘) it must follow that m∠ 4 =29^(∘). Supplementary angles have measures that sum to 180^(∘). Additionally, since we also know that ∠ 1 and ∠ 3 are vertical angles, we can write the following three equations. m∠ 4+m∠ 1&=180 m∠ 4+m∠ 3&=180 m∠ 1&=m∠ 3 By solving the first equation for m∠ 1, we also figure out the measure of ∠ 3.

m∠ 4+m∠ 1=180

Substitute

m∠ 4= 29

29+m∠ 1=180

SubEqn

LHS-29=RHS-29

m∠ 1=151

Let's summarize what we have found. m∠ 1&=151^(∘) m∠ 2&=29^(∘) m∠ 3&=151^(∘)