Question

1. $m=\frac{?}{?}$

$n=\frac{?}{?}$
$p=\frac{?}{?}$
$q=\frac{?}{?}$
$r=\frac{?}{?}$
$s=\frac{?}{?}$
$t=\frac{?}{?}$
$u=\frac{?}{?}$
construct each figure.

Explanation:

Step1: Find \(m\)

Using the fact that the sum of angles in a triangle - like part (with \(m\) and \(60^{\circ}\) and right - angle \(90^{\circ}\)): \(m+60^{\circ}+90^{\circ}=180^{\circ}\). So \(m = 180^{\circ}-(90^{\circ}+60^{\circ})=30^{\circ}\).

Step2: Find \(n\)

Using the property of vertical angles. The angle adjacent to \(112^{\circ}\) is \(180 - 112=68^{\circ}\). Then, for the triangle (with \(n\), \(40^{\circ}\) and right - angle \(90^{\circ}\)): \(n+40^{\circ}+90^{\circ}=180^{\circ}\). So \(n=180^{\circ}-(90^{\circ}+40^{\circ}) = 50^{\circ}\).

Step3: Find \(p\)

\(p\) and \(112^{\circ}\) are vertical angles. So \(p = 112^{\circ}\).

Step4: Find \(q\)

Using the property of supplementary angles. \(q+70^{\circ}=180^{\circ}\). So \(q = 180^{\circ}-70^{\circ}=110^{\circ}\).

Step5: Find \(r\)

\(r\) and \(q\) are vertical angles. So \(r=q = 110^{\circ}\).

Step6: Find \(s\)

\(s\) and \(70^{\circ}\) are vertical angles. So \(s = 70^{\circ}\).

Step7: Find \(t\)

\(t\) and \(n\) are vertical angles. So \(t=n = 50^{\circ}\).

Step8: Find \(u\)

Using the property of vertical angles (or the fact that it is a right - angle in the right - angled figure). \(u = 90^{\circ}\).

Answer:

\(m = 30^{\circ}\), \(n = 50^{\circ}\), \(p = 112^{\circ}\), \(q = 110^{\circ}\), \(r = 110^{\circ}\), \(s = 70^{\circ}\), \(t = 50^{\circ}\), \(u = 90^{\circ}\)