Question

Paragraph Proofs

Answer 1

For the proof here kindly check the attachment.

We are given that$r\parallel s$. Also, the transversal is shown. Let us take the first case, that of $\angle 1$ and $\angle 5$. Please note that all other proofs will follow in a similar manner.

Let us begin, please have a nice look at the diagram. We will see that $\angle 1$ and $\angle 3$ are vertically opposite angles. We know that vertically opposite angles are congruent. Thus, $\angle 1$ and $\angle 3$ are congruent angles.

$\angle 1$ = $\angle 3$

Now, we know that $\angle 3$ and $\angle 5$ are alternate interior angles. We also, know that alternate interior angles are equal too. Thus, we have:

$\angle 3$ = $\angle 5$

From the above arguments it is clear that:

$\angle 1$ = $\angle 3$ = $\angle 3$.

Thus, $\angle 1$ = $\angle 3$

We have proven the first instance. Please note that all other instances can be proved in a similar fashion.

For example, for $\angle 2$ and $\angle 6$ we can take $\angle 2$ and $\angle 4$ as vertically opposite angles thus making $\angle 2$ = $\angle 4$. Now, $\angle 4$ and $\angle 6$ are alternate interior angles and thus $\angle 4$ and $\angle 6$ are equal. Thus, we have $\angle 2$ and $\angle 6$ .

Answer 2

Answer:

Lines r and s are parallel as Corresponding Angles given. There are four pairs of corresponding angles: angle 1 and angle 5, angle 2 and angle 6, angle 3 and angle 7, and angle 4 and angle 8. Since r and s are parallel, the slope of r is equal to the slope of s. Since t is a straight line, the slope of t is the same at both intersections, by the definition of a straight line. Thus, the corresponding angles created at both intersections must have the same measure, since the difference of the slopes at each intersection is the same, and the intersections share a common line. So, corresponding angles must have equal measure. Therefore, by definition of congruent angles, corresponding angles are congruent: angle 1 is congruent to angle 5, angle 2 is congruent to angle 6, angle 3 is congruent to angle 7, and angle 4 is congruent to angle 8.