Exercise 66 Page 110

In this diagram, we see two parallel lines cut by two transversals.

Examining the diagram, we can identify a pair of alternate interior angles, a linear pair, and a pair of vertical angles. If we introduce a seventh angle, θ, we can also identify a pair of corresponding angles and a second linear pair.

Since the two lines are parallel, by the Alternate Interior Angles Theorem and the Corresponding Angles Theorem we can say that these angle pairs have equal measures.

We also know that a linear pair is supplementary, and that vertical angles have equal measures due to the Vertical Angles Theorem. With this information we can write a few equations. &m∠ θ =62^(∘) &m∠ d = m∠ c &m∠ b = m∠ a &m∠ c + 148 ^(∘) =180^(∘) &m∠ θ+ m∠ a =180^(∘) By solving for m∠ c and m∠ a in the last two equations we will have enough information to find all angle measures.

m∠ c + 148 ^(∘) =180^(∘)

SubEqn

LHS-148 ^(∘)=RHS-148 ^(∘)

m∠ c =32^(∘)

Since we know the measure of ∠ θ, we can find the measure of ∠ a.

m∠ θ+ m∠ a =180^(∘)

Substitute

m∠ θ= 62^(∘)

62^(∘)+ m∠ a =180^(∘)

SubEqn

LHS-62^(∘)=RHS-62^(∘)

m∠ a =118^(∘)

We know that m∠ a = 118^(∘) and m∠ c = 32^(∘), so we can find the remaining angles. &m∠ d = 32^(∘) &m∠ b = 118^(∘) &m∠ c = 32^(∘) &m∠ a = 118^(∘) Let's summarize what we have found in a table. |c|c|c| [-0.9em] -2ptAngle -2pt & -2pt Measure -2pt & Justification [0.5em] [-0.9em] ∠ a & 118^(∘) & Linear Pair Postulate [0.5em] [-0.9em] ∠ b & 118^(∘) & -2pt Vertical Angles Theorem -2pt [0.5em] [-0.9em] ∠ c & 32^(∘) & Linear Pair Postulat [0.5em] [-0.9em] ∠ d & 32^(∘) & Alternate Interior [-0.5em] Angles Theorem