Question

1. determine the value of x in each diagram.

a. b.
c. d.

Explanation:

Step1: Use the exterior angle theorem for triangle \( \triangle HJI \)

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \( \triangle HJI \), the exterior angle \( \angle HJK=81^{\circ} \), and the two non - adjacent interior angles are \( 33^{\circ} \) and \( 2x^{\circ} \). So, \( 33 + 2x=81 \).
Subtract \( 33 \) from both sides: \( 2x=81 - 33 \).
\( 2x = 48 \).
Divide both sides by \( 2 \): \( x=\frac{48}{2}=24 \).

Step2: Use the exterior angle theorem for triangle \( \triangle UTV \)

In \( \triangle UTV \), the exterior angle \( \angle UV S\) (where \( \angle UV S=(x + 8)^{\circ} \)) is equal to the sum of the two non - adjacent interior angles \( 90^{\circ} \) and \( 64^{\circ} \).
So, \( 90+64=x + 8 \).
\( 154=x + 8 \).
Subtract \( 8 \) from both sides: \( x=154 - 8=146 \).

Step3: Use the exterior angle property (linear pairs and exterior angle theorem)

First, find the interior angle at \( J \): \( 180-132 = 48^{\circ} \).
Find the interior angle at \( L \): \( 180 - 112=68^{\circ} \).
In \( \triangle JKL \), using the exterior angle theorem (or the angle - sum property of a triangle: \( 48+(2x + 4)=68 \)).
\( 2x+4=68 - 48 \).
\( 2x+4 = 20 \).
Subtract \( 4 \) from both sides: \( 2x=20 - 4=16 \).
Divide by \( 2 \): \( x = 8 \).

Step4: Use the exterior angle theorem for triangle \( \triangle DEF \)

The exterior angle \( \angle GFE = 90^{\circ} \), and the two non - adjacent interior angles are \( (3x + 2)^{\circ} \) and \( 46^{\circ} \).
So, \( 3x+2+46=90 \).
\( 3x+48=90 \).
Subtract \( 48 \) from both sides: \( 3x=90 - 48=42 \).
Divide by \( 3 \): \( x=\frac{42}{3}=14 \).

Answer:

a. \( x = 24 \)
b. \( x = 146 \)
c. \( x = 8 \)
d. \( x = 14 \)