Question

for each part below, use the figure to fill in the blank. if necessary, you may learn what the markings on a figure indicate. (a) find m ∠ m. m n p m ∠ m = \square ° (b) find m ∠ q. r 8 30° 8 q s m ∠ q = \square °

Explanation:

Part (a)

Step 1: Identify Triangle Type

Triangle \( MNP \) has all three sides marked as equal (tick marks), so it's equilateral? Wait, no, wait—the markings: \( MN \) has one tick, \( MP \) and \( NP \) have one tick? Wait, no, looking at the figure: \( MN \) has one tick, \( MP \) and \( NP \) have one tick each? Wait, no, maybe it's isosceles? Wait, no, if all three sides have the same tick, it's equilateral. Wait, no, the figure: \( M \) to \( N \): one tick, \( M \) to \( P \): one tick, \( N \) to \( P \): one tick? Wait, no, maybe the triangle is equilateral, so all angles are \( 60^\circ \)? Wait, no, maybe I misread. Wait, the triangle \( MNP \): sides \( MP \) and \( NP \) have the same tick, and \( MN \) has the same tick? So all three sides equal, so equilateral triangle. Therefore, all angles are \( 60^\circ \). So \( m\angle M = 60^\circ \).

Wait, maybe it's isosceles? Wait, no, if all three sides are marked with the same tick, it's equilateral. So each angle is \( 60^\circ \). So \( m\angle M = 60 \).

Part (b)

Step 1: Identify Triangle Type

Triangle \( RQS \) (wait, \( RQS \)? No, \( RQS \) is \( RQ = RS = 8 \), so it's isosceles with \( RQ = RS \). Therefore, angles opposite those sides are equal: \( \angle Q = \angle S \).

Step 2: Sum of Angles in Triangle

Sum of angles in a triangle is \( 180^\circ \). Given \( \angle R = 30^\circ \), so \( \angle Q + \angle S + \angle R = 180^\circ \). Since \( \angle Q = \angle S \), let \( \angle Q = x \), then \( 2x + 30^\circ = 180^\circ \).

Step 3: Solve for \( x \)

\( 2x = 180^\circ - 30^\circ = 150^\circ \), so \( x = \frac{150^\circ}{2} = 75^\circ \). Therefore, \( m\angle Q = 75^\circ \).

Answer:

(a) \( m\angle M = \boxed{60}^\circ \)
(b) \( m\angle Q = \boxed{75}^\circ \)