Question

1. find the value of each angle.

Explanation:

Step1: Find \(\angle1\)

Since there is a right - angle, \(\angle1 = 90^{\circ}\)

Step2: Find \(\angle2\)

Vertical angles are equal. \(\angle2+\angle1 = 180^{\circ}\) (linear pair). But we can also use the property of the sum of angles in a triangle. However, another way: \(\angle2=180^{\circ}-\angle1\). Since \(\angle1 = 90^{\circ}\), \(\angle2 = 90^{\circ}\)

Step3: Find \(\angle4\)

In the left - hand triangle (with \(\angle1 = 90^{\circ}\) and \(40^{\circ}\)), using the angle - sum property of a triangle (\(\angle1+\angle4 + 40^{\circ}=180^{\circ}\)). Substitute \(\angle1 = 90^{\circ}\), then \(90^{\circ}+\angle4+40^{\circ}=180^{\circ}\), \(\angle4=180^{\circ}-(90^{\circ} + 40^{\circ})=50^{\circ}\)

Step4: Find \(\angle3\)

\(\angle3=\angle4\) (vertical angles). So \(\angle3 = 50^{\circ}\)

Step5: Find \(\angle6\)

In the left - hand triangle (right - angled), \(\angle6+40^{\circ}=90^{\circ}\) (because \(\angle1 = 90^{\circ}\)), \(\angle6=90^{\circ}-40^{\circ}=50^{\circ}\)

Step6: Find \(\angle5\)

In the right - hand triangle (right - angled), \(\angle3+\angle5=90^{\circ}\) (angle - sum property of a right - angled triangle). Since \(\angle3 = 50^{\circ}\), \(\angle5=90^{\circ}-\angle3\), \(\angle5 = 40^{\circ}\)

Answer:

\(\angle1 = 90^{\circ}\), \(\angle2 = 90^{\circ}\), \(\angle3 = 50^{\circ}\), \(\angle4 = 50^{\circ}\), \(\angle5 = 40^{\circ}\), \(\angle6 = 50^{\circ}\)