Question

$overline{pr} cong overline{qs}$. complete the proof that $overline{qr} perp overline{pq}$.

(image of a quadrilateral with vertices r, s, p, q and diagonals intersecting at t, and a table with statement and reason columns:

1. $overline{pq} parallel overline{rs}$; reason: given
2. $overline{ps} parallel overline{qr}$; reason: given
3. $overline{pr} cong overline{qs}$; reason: given
4. $overline{pq} cong overline{rs}$; reason: given
5. $overline{qr} cong overline{qr}$; reason: parallelograms have congruent opposite sides
6. $\triangle pqr cong \triangle srq$; reason: reflexive property of congruence
7. $angle pqr cong angle qrs$; reason: sss
8. $mangle pqr + mangle qrs = 180^circ$; reason: cpctc
9. $mangle pqr + mangle pqr = 180^circ$; reason: same - side interior angles theorem
10. blank ; reason: substitution)

Explanation:

Step1: Simplify the equation from step 9

From step 9, we have \( m\angle PQR + m\angle PQR = 180^\circ \), which is \( 2m\angle PQR = 180^\circ \).

Step2: Solve for \( m\angle PQR \)

Divide both sides of the equation \( 2m\angle PQR = 180^\circ \) by 2: \( m\angle PQR=\frac{180^\circ}{2} = 90^\circ \).

Step3: Conclude the perpendicularity

If the measure of an angle \( \angle PQR \) is \( 90^\circ \), then the sides forming the angle, \( \overline{QR} \) and \( \overline{PQ} \), are perpendicular, so \( \overline{QR}\perp\overline{PQ} \).

Answer:

Statement 10: \( m\angle PQR = 90^\circ \) (or \( \overline{QR}\perp\overline{PQ} \))
Reason for statement 10: Division Property of Equality (for the angle measure) or Definition of Perpendicular Lines (since \( m\angle PQR = 90^\circ \))