Question

find the measure of each angle indicated.

1. 65° ?

57°
2)
40°
?
3)
20°
130° ?
4)
85°
? 50°
1
2
3
4
65 58 30 45 75 40 50

Explanation:

Step1: Use the triangle - angle sum theorem ($180^{\circ}$) for the first triangle

Let the unknown angle be $x$. For the first triangle, by the triangle - angle sum theorem ($\text{sum of angles in a triangle}=180^{\circ}$), we have $x + 65^{\circ}+57^{\circ}=180^{\circ}$.
So, $x=180^{\circ}-(65^{\circ} + 57^{\circ})$.
Calculate $65^{\circ}+57^{\circ}=122^{\circ}$, then $x = 180^{\circ}-122^{\circ}=58^{\circ}$.

Step2: Use the right - triangle angle sum theorem for the second triangle

In a right - triangle (one angle is $90^{\circ}$), let the unknown angle be $y$. We know that $y+40^{\circ}+90^{\circ}=180^{\circ}$.
So, $y=180^{\circ}-(40^{\circ}+90^{\circ})$.
Calculate $40^{\circ}+90^{\circ}=130^{\circ}$, then $y = 180^{\circ}-130^{\circ}=50^{\circ}$.

Step3: Use the triangle - angle sum theorem for the third triangle

Let the unknown angle be $z$. For the third triangle, $z+20^{\circ}+130^{\circ}=180^{\circ}$.
So, $z=180^{\circ}-(20^{\circ}+130^{\circ})$.
Calculate $20^{\circ}+130^{\circ}=150^{\circ}$, then $z = 180^{\circ}-150^{\circ}=30^{\circ}$.

Step4: Use the triangle - angle sum theorem for the fourth triangle

Let the unknown angle be $w$. For the fourth triangle, $w + 85^{\circ}+50^{\circ}=180^{\circ}$.
So, $w=180^{\circ}-(85^{\circ}+50^{\circ})$.
Calculate $85^{\circ}+50^{\circ}=135^{\circ}$, then $w = 180^{\circ}-135^{\circ}=45^{\circ}$.

Answer:

1. $58^{\circ}$
2. $50^{\circ}$
3. $30^{\circ}$
4. $45^{\circ}$