Question

2/2 rectangles, rhombus, squares practice day

1. if pqrs is a rectangle, st = 12, and ( mangle prs = 23^circ ), find each measure.

a) ( sq = )
b) ( pr = )
c) ( mangle qpr = )
d) ( mangle psr = )
e) ( mangle sqr = )
f) ( mangle ptq = )

Explanation:

Step1: Find length of SQ

In a rectangle, diagonals bisect each other, so $SQ = 2 \times ST$.
$SQ = 2 \times 12 = 24$

Step2: Find length of PR

In a rectangle, diagonals are congruent, so $PR = SQ$.
$PR = 24$

Step3: Find $m\angle QPR$

Alternate interior angles are equal, so $m\angle QPR = m\angle PRS$.
$m\angle QPR = 23^\circ$

Step4: Find $m\angle PSR$

All angles in a rectangle are right angles.
$m\angle PSR = 90^\circ$

Step5: Find $m\angle SQR$

First, $m\angle SRQ = 90^\circ$, so $m\angle SQR = 90^\circ - m\angle PRS$.
$m\angle SQR = 90^\circ - 23^\circ = 67^\circ$

Step6: Find $m\angle PTQ$

$\angle PTR$ and $\angle PRS$ form a triangle, so $m\angle PTR = 180^\circ - 2\times23^\circ = 134^\circ$. $\angle PTQ$ is supplementary to $\angle PTR$.
$m\angle PTQ = 180^\circ - 134^\circ = 46^\circ$

Answer:

a) $SQ = 24$
b) $PR = 24$
c) $m\angle QPR = 23^\circ$
d) $m\angle PSR = 90^\circ$
e) $m\angle SQR = 67^\circ$
f) $m\angle PTQ = 46^\circ$