Question

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

\\(\overline{qp} \cong \overline{qr}\\)
\\(\overline{pm} \cong \overline{mr}\\)
\\(\overline{qr} \cong \overline{rs}\\)
\\(\angle pqr \cong \angle psr\\)
\\(\angle qps \cong \angle qrs\\)

Explanation:

Analyze the definition of a kite

Using the Kite Properties knowledge point

• A kite is a quadrilateral with two distinct pairs of adjacent congruent sides.
• In the given diagram of kite \(PQRS\), the adjacent sides that are congruent are \(\overline{QP} \cong \overline{QR}\) and \(\overline{SP} \cong \overline{SR}\) (indicated by the tick marks on \(\overline{SP}\) and \(\overline{SR}\)).
• Therefore, the statement \(\overline{QP} \cong \overline{QR}\) must be true.

Analyze the diagonals of a kite

Using the Kite Properties knowledge point

• The diagonals of a kite are perpendicular, and the main diagonal (which is the line of symmetry, \(\overline{QS}\)) bisects the cross diagonal (\(\overline{PR}\)).
• Since \(\overline{QS}\) bisects \(\overline{PR}\) at point \(M\), the segment \(\overline{PM}\) is congruent to \(\overline{MR}\).
• Therefore, the statement \(\overline{PM} \cong \overline{MR}\) must be true.

Analyze the angles of a kite

Using the Kite Properties knowledge point

• A kite has exactly one pair of opposite angles that are congruent. These are the angles between the non-congruent adjacent sides.
• In kite \(PQRS\), the congruent opposite angles are \(\angle QPS\) and \(\angle QRS\).
• The other pair of opposite angles, \(\angle PQR\) and \(\angle PSR\), are generally not congruent unless the kite is also a rhombus.
• Therefore, the statement \(\angle QPS \cong \angle QRS\) must be true.

Answer:

• \(\overline{QP} \cong \overline{QR}\) (Correct answer)
• \(\overline{PM} \cong \overline{MR}\) (Correct answer)
• \(\overline{QR} \cong \overline{RS}\)
• \(\angle PQR \cong \angle PSR\)
• \(\angle QPS \cong \angle QRS\) (Correct answer)