Question

solve for the missing angle measures.
a.
b.

Explanation:

Step1: Analyze part a

• The left - hand triangle is equilateral (all sides equal), so all its angles are \(60^{\circ}\).
• In the right - hand part, we have a right - angled triangle (one angle \(90^{\circ}\)).
• For \(x\):
• Since the left - hand triangle is equilateral (\(60^{\circ}\) angles) and we consider the angle relationship. \(x = 60\div2=30^{\circ}\) (because of the side - equal relationship and angle bisecting - like property in the combined figure).
• For \(y\):
• Using the angle sum property of a quadrilateral (sum of interior angles of a quadrilateral is \(360^{\circ}\)).
• \(y=360-(90 + 60+90)=120^{\circ}\) (the angles \(90^{\circ}\) comes from the right - angle, \(60^{\circ}\) from the equilateral triangle part).

Step2: Analyze part b

• The left - hand triangle has two equal sides (isosceles triangle). Let's first find the base - angle of the left - hand isosceles triangle.
• The sum of angles in a triangle is \(180^{\circ}\). For the left - hand isosceles triangle with one angle \(50^{\circ}\), the base - angles are \(\frac{180 - 50}{2}=65^{\circ}\).
• For \(m\):
• \(m = 180 - 65=115^{\circ}\) (linear - pair relationship).
• For \(n\):
• The right - hand triangle has two equal sides (isosceles triangle). The angle adjacent to \(m\) is \(65^{\circ}\) (vertical - angle or angle - relationship in the figure).
• Using the sum of angles in a triangle (\(180^{\circ}\)) for the right - hand isosceles triangle. \(n=\frac{180 - 65}{2}=32.5^{\circ}\)

Answer:

a. \(x = 30^{\circ}\), \(y = 120^{\circ}\)
b. \(m = 115^{\circ}\), \(n = 32.5^{\circ}\)