Question

1. which statements (if any) are true about the diagram? select all that apply: a. point q is the midpoint of line pt. b. ∠pqs and ∠sqt are supplementary. c. ∠pqt is a straight angle. d. none of these 9. which statements (if any) are true about the diagram? select all that apply: a. ray eb is an angle bisector. b. point e is the midpoint of line segment ac. c. ∠aeb = 90° d. none of these

Explanation:

8.

Step1: Analyze option A

There is no information indicating \( Q \) divides \( PT \) into two equal parts. So, \( Q \) is not necessarily the mid - point of \( PT \).

Step2: Analyze option B

\(\angle PQS+\angle SQT=\angle PQT\). Since \( P, Q, T \) are collinear, \(\angle PQT = 180^{\circ}\). By the definition of supplementary angles (two angles whose sum is \( 180^{\circ}\)), \(\angle PQS\) and \(\angle SQT\) are supplementary.

Step3: Analyze option C

Since \( P, Q, T \) are collinear, \(\angle PQT\) is a straight angle (\(180^{\circ}\)).

9.

Step1: Analyze option A

There is no information that \( EB \) divides an angle into two equal non - right angles. A right - angle bisector would divide a \(180^{\circ}\) angle (if we consider \( \angle AEC\)) into two \(90^{\circ}\) angles, but the term "angle bisector" usually refers to non - right angle division in a more general sense when not specified. Here, \( \angle AEB = 90^{\circ}\) and \( \angle BEC=90^{\circ}\), but it's a special case (perpendicular).

Step2: Analyze option B

There is no information indicating \( E \) divides \( AC \) into two equal line segments.

Step3: Analyze option C

Since \( EB\perp AC\) (the right - angle symbol is shown), \(\angle AEB = 90^{\circ}\).

Answer:

1. B. \(\angle PQS\) and \(\angle SQT\) are supplementary; C. \(\angle PQT\) is a straight angle.
2. C. \(\angle AEB = 90^{\circ}\)