Question

1. determine the value of x in each diagram.

a.
b.
c.
d.

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. In part a, the angle with measure $81^{\circ}$ and the angle composed of $30^{\circ}+2x^{\circ}$ are vertical angles. So, we set up the equation $30 + 2x=81$.

Step2: Solve the equation for x

Subtract 30 from both sides: $2x=81 - 30=51$. Then divide both sides by 2: $x=\frac{51}{2}=25.5$.

Step3: For part b, use the exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. The exterior angle is $(x + 8)^{\circ}$, and the two non - adjacent interior angles are $90^{\circ}$ and $64^{\circ}$. So, $x + 8=90+64$.

Step4: Solve for x in part b

$x+8 = 154$, subtract 8 from both sides: $x=154 - 8 = 146$.

Step5: For part c, use the exterior - angle property

The exterior angle of the triangle at the left is $132^{\circ}$, and the two non - adjacent interior angles are $(2x + 4)^{\circ}$ and the angle whose exterior angle is $112^{\circ}$. The interior angle corresponding to the $112^{\circ}$ exterior angle is $180 - 112=68^{\circ}$. So, $132=(2x + 4)+68$.

Step6: Solve the equation in part c

First, simplify the right - hand side: $(2x + 4)+68=2x+72$. Then we have $132=2x + 72$. Subtract 72 from both sides: $2x=132 - 72 = 60$. Divide both sides by 2: $x = 30$.

Step7: For part d, use the angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, $(3x+2)+46 + 90=180$.

Step8: Solve for x in part d

Combine like terms: $3x+2+46 + 90=3x + 138$. Then $3x+138=180$. Subtract 138 from both sides: $3x=180 - 138 = 42$. Divide both sides by 3: $x = 14$.

Answer:

a. $x = 25.5$
b. $x = 146$
c. $x = 30$
d. $x = 14$