Question

1. find the measures of angles 1, 2, and 3 given that trap is a trapezoid with (overline{tr}paralleloverline{pa}). justify your answers carefully.

Explanation:

Step1: Use angle - sum property of right - triangle for angle 1

In right - triangle $TPR$, one angle is $90^{\circ}$ and another is $30^{\circ}$. The sum of the interior angles of a triangle is $180^{\circ}$. Let $\angle1=x$. Then $x + 90^{\circ}+30^{\circ}=180^{\circ}$. Solving for $x$, we get $x=\angle1 = 60^{\circ}$.

Step2: Use alternate - interior angles for angle 2

Since $\overline{TR}\parallel\overline{PA}$, $\angle2$ and the $30^{\circ}$ angle are alternate - interior angles. Alternate - interior angles formed by parallel lines and a transversal are equal. So $\angle2 = 30^{\circ}$.

Step3: Use angle - sum property of triangle for angle 3

In $\triangle RPA$, we know one angle is $40^{\circ}$ and from step 2, the angle at $P$ (adjacent to the $30^{\circ}$ angle) and the angle at $R$ (angle 3) and the $40^{\circ}$ angle sum to $180^{\circ}$. Let $\angle3 = y$. Then $y+40^{\circ}+30^{\circ}=180^{\circ}$. Solving for $y$, we get $y=\angle3 = 110^{\circ}$.

Answer:

$\angle1 = 60^{\circ}$, $\angle2 = 30^{\circ}$, $\angle3 = 110^{\circ}$