Question

(score for question 1: ___ of 6 points)

1. write a paragraph proof for the following conjecture

given: \\( \overline { q s } \\) bisects \\( \angle p q r \\)
\\( m \angle p q s = 45 ^ { \circ } \\)
prove: \\( p q r \\) is a right triangle
answer:

Explanation:

Step1: Use the angle - bisector definition

Since \( \overline{QS}\) bisects \( \angle PQR\), by the definition of an angle bisector, \( \angle PQS=\angle SQR\). Given that \( m\angle PQS = 45^{\circ}\), then \( m\angle SQR=45^{\circ}\).

Step2: Calculate \( m\angle PQR\)

Using the angle - addition postulate \( m\angle PQR=m\angle PQS + m\angle SQR\). Substituting the values of \( m\angle PQS\) and \( m\angle SQR\) (both \( 45^{\circ}\)), we get \( m\angle PQR=45^{\circ}+ 45^{\circ}=90^{\circ}\).

Step3: Use the right - triangle definition

A triangle is a right triangle if one of its angles is a right angle (\( 90^{\circ}\)). Since \( \angle PQR = 90^{\circ}\) in \( \triangle PQR\), by the definition of a right triangle, \( \triangle PQR\) is a right triangle.

Answer:

Since \( \overline{QS}\) bisects \( \angle PQR\), \( \angle PQS=\angle SQR\). Given \( m\angle PQS = 45^{\circ}\), then \( m\angle SQR = 45^{\circ}\). By the angle - addition postulate \( m\angle PQR=m\angle PQS + m\angle SQR=45^{\circ}+45^{\circ}=90^{\circ}\). A triangle with a \( 90^{\circ}\) angle is a right triangle. So, \( \triangle PQR\) is a right triangle.