Question

If quadrilateral PQRS is a kite, which statements must be true? Check all that apply.

Answer 1

Answer: The true statements are,

QP ≅ QR

PM ≅ MR

∠QPS ≅ ∠QRS

Step-by-step explanation:

A kite, is a quadrilateral which has two pairs of congruent sides.

Also, In kite the major diagonal bisects the minor diagonal perpendicularly.

Here, PQRS is a kite, in which,

Sides PQ and PS are congruent  to sides QR and RS respectively,

That is, QP ≅ QR and PS ≅ RS

Since, QS ≅ QS ( Reflexive )

By SSS postulate of congruence,

Δ PQS ≅ Δ RQS

By CPCTC,

∠ QPR ≅ ∠ QPS

Also, QS is the major diagonal and PR is the minor diagonal,

⇒ QS bisects PR

⇒ PM ≅ MR ( Where M is the intersection point of these diagonals. )

⇒ First second and fifth options are correct.

Answer 2

A kite is a quadrilateral in which two disjoint pairs of consecutive sides are congruent (“disjoint pairs” means that one side can’t be used in both pairs).

Then:

• Two disjoint pairs of consecutive sides are congruent by definition - QP≅QR and QR is not congruent to RS (one side can’t be used in both pairs);
• One diagonal (segment QS, the main diagonal) is the perpendicular bisector of the other diagonal (segment PR, the cross diagonal), so PM≅MR;
• The opposite angles at the endpoints of the cross diagonal are congruent, thus ∠QPS≅∠QRS.
• ∠PQR is not congruent to ∠PSR, because they are not angles at the endpoints of the cross diagonal.

Answer: correct options are A, B and E.