Question

r is the midpoint of pt and qs.

1. check all that are true based on the given information and the diagram above.

$\boldsymbol{pr cong tr}$ (definition of bisect)
$\boldsymbol{pr cong tr}$ (definition of midpoint)
$\boldsymbol{pq cong ts}$ (parallel lines)
$\boldsymbol{qr cong sr}$ (definition of midpoint)
$\boldsymbol{angle p cong angle t}$ (alternate interior angles)
$\boldsymbol{angle prq cong angle trs}$ (vertical angles)
$\boldsymbol{angle pqr cong angle str}$ (vertical angles)

Explanation:

1. $PR \cong TR$: R is the midpoint of PT, so it splits PT into two congruent segments. This matches the definition of a midpoint (not bisect, which refers to splitting a segment/angle into two equal parts, but the specific term here for a segment's midpoint is the correct justification).
2. $QR \cong SR$: R is the midpoint of QS, so it splits QS into two congruent segments, which fits the definition of a midpoint.
3. $\angle PRQ \cong \angle TRS$: These are opposite angles formed by intersecting lines PT and QS, so they are vertical angles, which are congruent.
4. The other options are incorrect:

• The first option uses "definition of bisect" which is not the precise term for a midpoint's effect on a segment here.
• There is no information stating PQ and TS are parallel, so we cannot conclude $PQ \cong TS$.
• We cannot confirm $\angle P \cong \angle T$ as alternate interior angles because we don't know if PQ and TS are parallel.
• $\angle PQR$ and $\angle STR$ are not vertical angles.

Answer:

• PR ≅ TR (Definition of midpoint)
• QR ≅ SR (Definition of midpoint)
• ∠PRQ ≅ ∠TRS (Vertical angles)