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 angle q ).
( m angle q=square^{circ} )
(b) find ( m angle m ).
( m angle m=square^{circ} )

Explanation:

Step1: Use the property of isosceles triangle (a)

In triangle \(QRS\), \(QR = SR=10\), so it is an isosceles triangle. The sum of interior angles of a triangle is \(180^{\circ}\). Let \(m\angle Q=m\angle S = x\). Then \(x + x+110^{\circ}=180^{\circ}\), which simplifies to \(2x=180^{\circ}- 110^{\circ}=70^{\circ}\), so \(x = 35^{\circ}\).

Step2: Use the property of equilateral triangle (b)

In triangle \(PMN\), \(PM = PN=MN\) (marked equal), so it is an equilateral triangle. In an equilateral triangle, all angles are equal. Since the sum of interior angles of a triangle is \(180^{\circ}\), if \(m\angle M = y\), then \(y+y + y=180^{\circ}\), so \(y = 60^{\circ}\).

Answer:

(a) \(m\angle Q = 35^{\circ}\)
(b) \(m\angle M=60^{\circ}\)