Question

1. find x, y and z (click here for video #4

Explanation:

Step1: Find \( x \)

Use the triangle angle - sum theorem (\( 180^{\circ} \) in a triangle).
For the left - hand triangle: \( x=180^{\circ}-30^{\circ}-64^{\circ}=86^{\circ} \)

Step2: Find \( y \)

Since the sum of angles on a straight line is \( 180^{\circ} \), \( y = 180^{\circ}-x - z \). First, find \( z \).
For the right - hand triangle: \( z=180^{\circ}-20^{\circ}-(180^{\circ}-(30^{\circ} + 20^{\circ})-(64^{\circ}))\).
Another way: The sum of angles in the large triangle. Let's use the property that the sum of angles in a triangle is \( 180^{\circ} \).
The third angle of the large triangle (opposite to the vertex) is \( 30^{\circ}+20^{\circ}=50^{\circ} \).
The sum of the base angles of the large triangle is \( 180^{\circ}-50^{\circ}=130^{\circ} \).
We know one base angle is \( 64^{\circ} \), so \( z=130^{\circ}-64^{\circ}-y\). Wait, better use the right - hand triangle.
In the right - hand triangle: \( z = 180^{\circ}-20^{\circ}-(180^{\circ}-x)\) (because the non - adjacent angle to \( x \) in the adjacent triangle). Substitute \( x = 86^{\circ} \), the angle adjacent to \( x \) in the right - hand triangle is \( 180^{\circ}-86^{\circ}=94^{\circ} \). Then \( z=180^{\circ}-20^{\circ}-94^{\circ}=66^{\circ} \)

Step3: Find \( y \)

Since \( x + y+z=180^{\circ} \) (straight line), \( y=180^{\circ}-x - z \). Substitute \( x = 86^{\circ} \) and \( z = 66^{\circ} \), \( y=180^{\circ}-86^{\circ}-66^{\circ}=28^{\circ} \)

Answer:

\( x = 86^{\circ}\), \( y = 28^{\circ}\), \( z = 66^{\circ}\)