Chapter 2.6, Problem 17WE

a.

To determine

To complete: the proof that $\angle 2\cong \angle 5$

Answer to Problem 17WE

Statements	Reasons
1.$\overline{PQ}\perp \overline{QR}$, $\overline{PS}\perp \overline{SR}$ 2.$\angle 2$ is comp. to $\angle 1$;$\angle 5$ is comp. to $\angle 4$ 3.$\angle 1\cong \angle 4$ 4.$\angle 2\cong \angle 5$	1.Given$\overline{PQ}\perp \overline{QR}$ 2.$\overline{PS}\perp \overline{SR}$ 3.Given 4.$\angle 2,\angle 5$are comp. of the congruent angles $\angle 1,\angle 4$

Explanation of Solution

Given information: $\overline{PQ}\perp \overline{QR}$ , $\overline{PS}\perp \overline{SR}$ and $\angle 1\cong \angle 4$

Formula used: If two angles are complements to the congruent angles then the two angles are congruent.

  

1. For statement $\overline{PQ}\perp \overline{QR}$ , $\overline{PS}\perp \overline{SR}$, reason is Given
2. For statement 2:

As $\overline{PQ}\perp \overline{QR}$ ,

  $\begin{array}{l}\angle 1+\angle 2=90°\\ \Rightarrow \angle 2=90°-\angle 1\end{array}$

So, $\angle 2$ is comp. to $\angle 1$ as sum of two complementary angles is equal to $90°$ .

As $\overline{PS}\perp \overline{SR}$ ,

  $\begin{array}{l}\angle 4+\angle 5=90°\\ \Rightarrow \angle 5=90°-\angle 4\end{array}$

So, $\angle 5$ is comp. to $\angle 4$ as sum of two complementary angles is equal to $90°$ .

3. For statement 3, reason is Given
4. For statement 4:

As $\angle 2$ is comp. to $\angle 1$ , $\angle 5$ is comp. to $\angle 4$ and $\angle 1\cong \angle 4$ ,

  $\angle 2\cong \angle 5$

(If two angles are complements to the congruent angles then the two angles are congruent.)

Statements	Reasons
5.$\overline{PQ}\perp \overline{QR}$, $\overline{PS}\perp \overline{SR}$ 6.$\angle 2$ is comp. to $\angle 1$;$\angle 5$ is comp. to $\angle 4$ 7.$\angle 1\cong \angle 4$ 8.$\angle 2\cong \angle 5$	5.Given 6. $\overline{PQ}\perp \overline{QR}$   $\overline{PS}\perp \overline{SR}$ 7. Given 8. $\angle 2,\angle 5$ are comp. of the congruent angles   $\angle 1,\angle 4$

b.

To determine

To prove: $\angle 3\cong \angle 6$

Explanation of Solution

Formula used: If two angles are supplementary to the congruent angles then the two angles are congruent.

Proof:

  

  $\angle 2+\angle 3=180°$ (linear pair)

  $\angle 3=180°-\angle 2$

So, $\angle 3$ is supplementary to $\angle 2$

(as sum of two supplementary angles is $180°$ )

  $\angle 5+\angle 6=180°$ (linear pair)

  $\angle 6=180°-\angle 5$

So, $\angle 6$ is supplementary to $\angle 5$

(as sum of two supplementary angles is $180°$ )

Also, $\angle 2\cong \angle 5$ (from part a.)

So,

  $\angle 3\cong \angle 6$

(If two angles are supplementary to the congruent angles then the two angles are congruent.)