Question

1. find the measure of each required angle and give the reason for your answer. write all angles in the order that you solve them.

a)
angle | reason

Explanation:

Find Angle 3

$$ \angle 3 = 63^\circ \quad (\text{Vertical angles}) $$

Find Angle 1

$$ \angle 1 = 180^\circ - \angle 3 = 180^\circ - 63^\circ = 117^\circ \quad (\text{Consecutive interior angles}) $$

Find Angle 2

$$ \angle 2 = 180^\circ - (\angle 1 + 70^\circ) = 180^\circ - (117^\circ + 70^\circ) = -7^\circ \quad (\text{Invalid geometry}) $$

Alternative interpretation: The top line is straight, so:

$$ \angle 1 + 70^\circ + \angle 2 = 180^\circ $$

If \(\angle 1 = 117^\circ\), this is impossible. Let's re-evaluate the transversal.
The angle labeled \(63^\circ\) and \(\angle 1\) are alternate interior angles:

$$ \angle 1 = 63^\circ \quad (\text{Alternate interior angles}) $$

Then, \(\angle 3\) and the \(63^\circ\) angle are supplementary on a straight line:

$$ \angle 3 = 180^\circ - 63^\circ = 117^\circ \quad (\text{Angles on a straight line}) $$

Now, find \(\angle 2\) using the straight line at the top:

$$ \angle 2 = 180^\circ - (\angle 1 + 70^\circ) = 180^\circ - (63^\circ + 70^\circ) = 47^\circ \quad (\text{Angles on a straight line}) $$

Finally, find \(\angle 4\) using alternate interior angles with \(\angle 2\):

$$ \angle 4 = \angle 2 = 47^\circ \quad (\text{Alternate interior angles}) $$

Answer:

Angle	Reason
\(\angle 3 = 117^\circ\)	Angles on a straight line (supplementary)
\(\angle 2 = 47^\circ\)	Angles on a straight line (supplementary)
\(\angle 4 = 47^\circ\)	Alternate interior angles