Question

quadrilateral rpqs is a rectangle. label the missing measures. p q pq = 5 pt = t 8 ps = m∠sqp = 1 m∠1 = 30° r 6 s 45° 60° 90° 10 6 5 8

Explanation:

Step1: Recall rectangle properties

In a rectangle, opposite sides are equal. So $PQ = RS = 6$.

Step2: Recall diagonals property

The diagonals of a rectangle are equal and bisect each other. Given the diagonal length from the right - angled triangle formed with sides 6 and 8, using the Pythagorean theorem $d=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10$. Since the diagonals bisect each other, $PT=\frac{10}{2}=5$.

Step3: Opposite sides equality

$PS = RQ = 8$.

Step4: Angle in a rectangle

Each angle of a rectangle is $90^{\circ}$, so $m\angle SQP=90^{\circ}$.

Step5: Use angle - sum property

In right - triangle $RPS$, $\angle RSP = 30^{\circ}$, and in rectangle, $\angle RSP=\angle SQP$. In $\triangle RPS$, $m\angle1 = 60^{\circ}$ (because in right - triangle $RPS$ with one angle $30^{\circ}$ and right - angle $90^{\circ}$, the third angle $m\angle1=180^{\circ}-90^{\circ}-30^{\circ}=60^{\circ}$).

Answer:

$PQ = 6$
$PT = 5$
$PS = 8$
$m\angle SQP=90^{\circ}$
$m\angle1 = 60^{\circ}$