Question

if quadrilateral pqrs is a kite, which statements must be true? select three options

qp ≅ qr
pm ≅ mr
qr ≅ rs
∠pqr ≅ ∠psr
∠qps ≅ ∠qrs

Explanation:

Step1: Recall the properties of a kite

A kite has two pairs of adjacent sides that are equal. Also, the diagonals of a kite are perpendicular and one of the diagonals bisects the other.

Step2: Analyze each option

• For \(\overline{QP}\cong\overline{QR}\): In a kite, two pairs of adjacent sides are equal. If \(PQRS\) is a kite, \(QP = QR\) (assuming \(Q\) is a vertex where two adjacent equal - side pairs meet).
• For \(\overline{PM}\cong\overline{MR}\): The diagonal \(PR\) is bisected by the other diagonal \(QS\) (a property of a kite). So \(PM = MR\).
• For \(\angle PQR\cong\angle PSR\): The non - vertex angles (the angles between the unequal sides) of a kite are equal. Here \(\angle PQR\) and \(\angle PSR\) are non - vertex angles.

• For \(\overline{QR}\cong\overline{RS}\): In a kite, adjacent sides are equal. \(QR\) and \(RS\) are not adjacent sides in the general definition of a kite (a kite has two distinct pairs of adjacent sides equal).
• For \(\angle QPS\cong\angle QRS\): \(\angle QPS\) and \(\angle QRS\) are vertex angles. Vertex angles of a kite are not necessarily equal.

Answer:

\(\overline{QP}\cong\overline{QR}\), \(\overline{PM}\cong\overline{MR}\), \(\angle PQR\cong\angle PSR\)