Question

1. determine the value of x in each diagram.

a.
b.
c.
d.

Explanation:

Step1: Use exterior - angle property for part a

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In triangle $HIJ$, the exterior angle at $J$ is $81^{\circ}$, and the two non - adjacent interior angles are $33^{\circ}$ and $2x$. So, $81 = 33+2x$.

Step2: Solve the equation for $x$ in part a

Subtract 33 from both sides of the equation $81 = 33 + 2x$:
$81-33=2x$, which simplifies to $48 = 2x$. Then divide both sides by 2: $x=\frac{48}{2}=24$.

Step3: Use angle - sum property for part b

The sum of angles in a triangle is $180^{\circ}$. In right - triangle $UTV$, one angle is $90^{\circ}$ and another is $64^{\circ}$. The exterior angle at $V$ is $x + 8$. First, find the third angle in $\triangle UTV$. Let the third angle be $y$, then $y=180-(90 + 64)=26^{\circ}$. The exterior angle at $V$ is equal to the sum of the two non - adjacent interior angles of $\triangle UTV$. So, $x + 8=90+64$.

Step4: Solve the equation for $x$ in part b

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

Step5: Use exterior - angle property for part c

The exterior angle at $J$ is $132^{\circ}$ and the exterior angle at $L$ is $112^{\circ}$. For the triangle with sides $JK$ and $KL$, using the exterior - angle property, $132=(2x + 4)+(180 - 112)$.

Step6: Simplify the equation for part c

First, simplify the right - hand side: $132=(2x + 4)+68$. Then $132=2x+72$.

Step7: Solve the equation for $x$ in part c

Subtract 72 from both sides: $132 - 72=2x$, so $60 = 2x$. Divide both sides by 2: $x = 30$.

Step8: Use exterior - angle property for part d

The exterior angle at $F$ is $90^{\circ}$. In triangle $DEF$, using the exterior - angle property, $90=(3x + 2)+46$.

Step9: Solve the equation for $x$ in part d

Subtract 46 from both sides: $90-46=3x + 2$, so $44=3x + 2$. Then subtract 2 from both sides: $42=3x$. Divide both sides by 3: $x = 14$.

Answer:

a. $x = 24$
b. $x = 146$
c. $x = 30$
d. $x = 14$