Question

the quadrilateral shown is a rectangle. label the missing measures. m∠1 = m∠2 = m∠3 = m∠atb = m∠atc = 68° 136° 44° 22°

Explanation:

Step1: Recall rectangle properties

In a rectangle, the diagonals are congruent and bisect each other. So, $\triangle BTD$ is isosceles. Given $\angle 1 = 22^{\circ}$, then $\angle 2=\angle 1 = 22^{\circ}$.

Step2: Find $\angle 3$

In right - triangle $BCD$, $\angle BCD = 90^{\circ}$. Since $\angle 1=22^{\circ}$, then $\angle 3=90^{\circ}-\angle 1=90^{\circ} - 22^{\circ}=68^{\circ}$.

Step3: Find $\angle ATB$

$\angle ATB$ and $\angle DTC$ are vertical angles. $\angle DTC = 180^{\circ}- 2\angle 1=180^{\circ}-44^{\circ}=136^{\circ}$, so $\angle ATB = 136^{\circ}$.

Step4: Find $\angle ATC$

$\angle ATC$ and $\angle ATB$ are supplementary. So $\angle ATC=180^{\circ}-\angle ATB = 44^{\circ}$.

Answer:

$m\angle1 = 22^{\circ}$
$m\angle2 = 22^{\circ}$
$m\angle3 = 68^{\circ}$
$m\angle ATB = 136^{\circ}$
$m\angle ATC = 44^{\circ}$