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 ( mangle q ). (b) find ( mangle m ).

Explanation:

Step1: Find \( m\angle Q \) (Isosceles triangle property)

In \(\triangle SRQ\), \(SR = QR = 4\). So, \(\triangle SRQ\) is isosceles. Let \(m\angle Q=m\angle S = x\).
By the angle - sum property of a triangle (\(m\angle S+m\angle Q + m\angle R=180^{\circ}\)), we have \(x + x+100^{\circ}=180^{\circ}\).
Simplify the equation: \(2x=180^{\circ}- 100^{\circ}=80^{\circ}\).
Solve for \(x\): \(x = 40^{\circ}\). So, \(m\angle Q = 40^{\circ}\).

Step2: Find \(m\angle M\) (Equilateral triangle property)

In \(\triangle NPM\), all the sides are marked as equal. So, \(\triangle NPM\) is equilateral.
By the angle - sum property of a triangle (\(m\angle N+m\angle M + m\angle P=180^{\circ}\)) and in an equilateral triangle all angles are equal. Let \(m\angle N=m\angle M = m\angle P=y\).
Then \(y + y + y=180^{\circ}\), \(3y = 180^{\circ}\).
Solve for \(y\): \(y = 60^{\circ}\). So, \(m\angle M=60^{\circ}\).

Answer:

(a) \(40\)
(b) \(60\)