Question

problem 3 finding measures of angles
what are the measures of ∠3 and ∠4? which theorem or postulate justifies each answer?
since , m∠3 = by the
since , by the
so, m∠4 =
got it? 3. use the diagram in problem 3. what is the measure of each angle? justify each answer.
a. ∠1
b. ∠2
c. ∠5
d. ∠6
e. ∠7
f. ∠8

Explanation:

Step1: Identify angle - angle relationships

$\angle3$ and the $105^{\circ}$ angle are vertical angles.

Step2: Apply vertical - angles theorem

Vertical angles are congruent. So $m\angle3 = 105^{\circ}$ by the vertical - angles theorem.

Step3: Identify another angle - angle relationship

$\angle4$ and the $105^{\circ}$ angle are corresponding angles.

Step4: Apply corresponding - angles postulate

If two parallel lines are cut by a transversal, corresponding angles are congruent. So $m\angle4=105^{\circ}$ by the corresponding - angles postulate.

For the "Got It?" part:
a. $\angle1$ and $\angle4$ are vertical angles. So $m\angle1 = 105^{\circ}$ by the vertical - angles theorem.
b. $\angle2$ and $\angle1$ are a linear pair. Since a linear pair of angles is supplementary ($m\angle2 + m\angle1=180^{\circ}$), $m\angle2=180 - 105=75^{\circ}$ by the linear - pair postulate.
c. $\angle5$ and $\angle2$ are vertical angles. So $m\angle5 = 75^{\circ}$ by the vertical - angles theorem.
d. $\angle6$ and $\angle4$ are vertical angles. So $m\angle6 = 105^{\circ}$ by the vertical - angles theorem.
e. $\angle7$ and the $105^{\circ}$ angle are vertical angles. So $m\angle7 = 105^{\circ}$ by the vertical - angles theorem.
f. $\angle8$ and $\angle1$ are vertical angles. So $m\angle8 = 105^{\circ}$ by the vertical - angles theorem.

Answer:

$m\angle3 = 105^{\circ}$ (vertical - angles theorem), $m\angle4 = 105^{\circ}$ (corresponding - angles postulate)
a. $m\angle1 = 105^{\circ}$ (vertical - angles theorem)
b. $m\angle2 = 75^{\circ}$ (linear - pair postulate)
c. $m\angle5 = 75^{\circ}$ (vertical - angles theorem)
d. $m\angle6 = 105^{\circ}$ (vertical - angles theorem)
e. $m\angle7 = 105^{\circ}$ (vertical - angles theorem)
f. $m\angle8 = 105^{\circ}$ (vertical - angles theorem)