Question

find the value of each variable.
10
11
12
x=

x=

x=

y=

y=

y=

z=

Explanation:

Problem 10

Step1: Find base angles of large isosceles triangle

The sum of angles in a triangle is \(180^{\circ}\). For the large isosceles triangle with vertex angle \(30^{\circ}\), let the base angles be \(B\). Then \(B=\frac{180 - 30}{2}=75^{\circ}\) (since in an isosceles triangle, base angles are equal).

Step2: Find \(x\)

The small right - angled isosceles triangle (because two sides are equal) has angles \(90^{\circ}, 45^{\circ},45^{\circ}\). \(x = 75-45=30^{\circ}\)

Step3: Find \(y\)

\(y = 45^{\circ}\) (from the right - angled isosceles triangle)

Step1: Use properties of isosceles triangles

The upper triangle is isosceles (two equal sides). Let the base angles of the upper isosceles triangle be \(A\). The lower triangle is also isosceles (two equal sides).
The angle adjacent to \(50^{\circ}\) in the upper triangle: \(180 - 50=130^{\circ}\). Since the upper triangle is isosceles, \(x=\frac{180 - 130}{2}=25^{\circ}\)

Step2: Use angle - sum property for \(y\)

For the lower isosceles triangle with vertex angle \(60^{\circ}\), \(y=\frac{180 - 60}{2}=60^{\circ}\) (because it is an isosceles triangle, base angles are equal)

Step1: Find \(x\)

The left - hand triangle is isosceles (two equal sides). So \(x = 60^{\circ}\) (equilateral triangle property as one angle is \(60^{\circ}\) and two sides are equal)

Step2: Find \(y\)

The middle triangle: using the property of parallel lines (if we consider the sides) and isosceles triangle. \(y = 60^{\circ}\)

Step3: Find \(z\)

For the exterior angle \(z\), using the angle - sum property of the triangle. \(z=60 + 60=120^{\circ}\) (exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles)

Answer:

\(x = 30^{\circ}\), \(y = 45^{\circ}\)

Problem 11