Question

find the measure of the numbered angle.
m∠6 = □°

Explanation:

Step1: Find the measure of ∠1

In a right - angled triangle, the sum of angles is \(180^{\circ}\). Given one angle is \(40^{\circ}\) and another is \(90^{\circ}\).
\(\angle1=180^{\circ}-(40^{\circ} + 90^{\circ})=50^{\circ}\)

Step2: Use vertical - angle property

Since \(\angle1\) and \(\angle3\) are vertical angles, \(\angle3=\angle1 = 50^{\circ}\)

Step3: Find the measure of ∠5

In the right - angled triangle with \(\angle3\), \(\angle5=180^{\circ}-(90^{\circ}+\angle3)\)
Substitute \(\angle3 = 50^{\circ}\), \(\angle5=180^{\circ}-(90^{\circ}+50^{\circ}) = 40^{\circ}\)

Step4: Use the linear - pair property

\(\angle6\) and \(\angle5\) form a linear pair (\(\angle6+\angle5 = 180^{\circ}\))
\(\angle6=180^{\circ}-\angle5\)
Substitute \(\angle5 = 40^{\circ}\), \(\angle6=140^{\circ}\)

Answer:

\(140\)