Question

free response questions - answer each of the following questions. show the work you do finding your final answer.

1. points p, q, r, and s are shown below. use the diagram to answer the following questions.

(a) draw \\( \angle pqr \\).
(b) what type of angle does \\( \angle pqr \\) appear to be?
(c) draw ray \\( \overrightarrow{qs} \\).
(d) the measure of \\( \angle pqs \\) is \\( 17x + 11 \\) and the measure of \\( \angle sqr \\) is \\( 21x - 1 \\). if \\( x = 3 \\), determine if \\( \overrightarrow{qs} \\) bisects \\( \angle pqr \\). show or explain how you arrived at your answer.

Explanation:

Step1: Calculate the measure of ∠PQS

Substitute \(x = 3\) into \(17x + 11\).
\(17\times3+11=51 + 11=62\)

Step2: Calculate the measure of ∠SQR

Substitute \(x = 3\) into \(21x - 1\).
\(21\times3-1=63 - 1=62\)

Step3: Check if ∠PQS = ∠SQR

Since \(∠PQS=62^{\circ}\) and \(∠SQR = 62^{\circ}\), by the definition of an angle - bisector (a ray that divides an angle into two equal - measure angles), \(\overrightarrow{QS}\) bisects \(∠PQR\).

Answer:

\(\overrightarrow{QS}\) bisects \(∠PQR\) because when \(x = 3\), \(∠PQS=17x + 11=17\times3+11 = 62\) and \(∠SQR=21x - 1=21\times3-1 = 62\), so \(∠PQS = ∠SQR\).