Question

7 x=____
find the measure of x, y, and z
triangle is not drawn to scale
y=____
z=____
8 x=____
find the value of x and y
y=____

Explanation:

7.

Step1: Find \(x\)

Since \(135^{\circ}\) and \(x\) are supplementary angles (they form a linear pair), we use the formula \(x + 135^{\circ}=180^{\circ}\).

$$x=180^{\circ}- 135^{\circ}=45^{\circ}$$

Step2: Find \(y\)

The triangle is isosceles (two sides are equal), so the base - angles are equal. Using the angle - sum property of a triangle (\(x + y+ \text{base - angle}=180^{\circ}\), and the base - angle adjacent to \(x\) is \(45^{\circ}\)). The sum of angles in a triangle is \(180^{\circ}\), so \(y + 2\times45^{\circ}=180^{\circ}\).

$$y = 90^{\circ}$$

Step3: Find \(z\)

\(z\) and the non - \(x\) base - angle of the triangle are supplementary. The non - \(x\) base - angle is \(45^{\circ}\), so \(z+45^{\circ}=180^{\circ}\)

$$z = 135^{\circ}$$

8.

Step1: Find \(x\)

The triangle with \(x\) is equilateral (all sides are equal), so all angles are equal. Using the angle - sum property of a triangle (\(x + x+60^{\circ}=180^{\circ}\), or since it's equilateral \(x = 60^{\circ}\))

$$x = 60^{\circ}$$

Step2: Find \(y\)

Using the exterior - angle property (the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). Consider the triangle formed by the two smaller triangles. The angle adjacent to \(60^{\circ}\) in the equilateral - like part is \(60^{\circ}\), and for the other triangle, if we assume some basic triangle properties (alternatively, if we consider parallel - line or triangle congruence/ similarity in a simple case). In a simple two - triangle case where one is equilateral and the other is a smaller triangle attached, and using the fact that the line with equal segments can imply some angle relationships. If we consider the angle \(y\) and the \(60^{\circ}\) angle in a way that \(y\) is half of \(60^{\circ}\) (using the property of angle bisector or a simple isosceles - triangle - related exterior - angle property in a combined - triangle figure)

$$y = 30^{\circ}$$

Answer:

1. \(x = 45^{\circ}\), \(y=90^{\circ}\), \(z = 135^{\circ}\)
2. \(x = 60^{\circ}\), \(y = 30^{\circ}\)