Question

1. t 5x + 10 58° s 11x + 2 u a 11) c 46° -1 + 8x d 18x + 5 b w 12) c 10x - 10 a 12x - 4 q 30° b 13) c 15x + 5 120° a d 22x + 4 b 14) d 9x - 2 b 20x + 5 v 40° c find the measure of the angle indicated.

Explanation:

Step1: Recall exterior - angle property of a triangle

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Solve problem 10

In triangle \(TSU\), the exterior angle \(\angle TAU=11x + 2\), and the non - adjacent interior angles are \(5x+10\) and \(58^{\circ}\). So, \(11x + 2=(5x + 10)+58\).
\[

$$\begin{align*} 11x+2&=5x + 68\\ 11x-5x&=68 - 2\\ 6x&=66\\ x&=11 \end{align*}$$

\]
The angle \(\angle T = 5x+10=5\times11 + 10=65^{\circ}\).

Step3: Solve problem 11

In triangle \(BCD\), the exterior angle \(\angle CBW = 18x+5\), and the non - adjacent interior angles are \(46^{\circ}\) and \(- 1+8x\). So, \(18x+5=46+( - 1+8x)\).
\[

$$\begin{align*} 18x+5&=46-1 + 8x\\ 18x-8x&=45 - 5\\ 10x&=40\\ x&=4 \end{align*}$$

\]
The angle \(\angle D=-1 + 8x=-1+8\times4 = 31^{\circ}\).

Step4: Solve problem 12

The exterior angle \(\angle CAQ=12x - 4\), and the non - adjacent interior angles are \(10x-10\) and \(30^{\circ}\). So, \(12x - 4=(10x-10)+30\).
\[

$$\begin{align*} 12x-4&=10x + 20\\ 12x-10x&=20 + 4\\ 2x&=24\\ x&=12 \end{align*}$$

\]
The angle \(\angle CAB=12x-4=12\times12-4 = 140^{\circ}\).

Step5: Solve problem 13

The exterior angle \(\angle ADB = 120^{\circ}\), and the non - adjacent interior angles are \(15x+5\) and \(22x + 4\). So, \(120=(15x+5)+(22x + 4)\).
\[

$$\begin{align*} 120&=37x+9\\ 37x&=120 - 9\\ 37x&=111\\ x&=3 \end{align*}$$

\]
The angle \(\angle C=15x+5=15\times3+5 = 50^{\circ}\).

Step6: Solve problem 14

The exterior angle \(\angle DBV=20x + 5\), and the non - adjacent interior angles are \(9x-2\) and \(40^{\circ}\). So, \(20x + 5=(9x-2)+40\).
\[

$$\begin{align*} 20x+5&=9x + 38\\ 20x-9x&=38 - 5\\ 11x&=33\\ x&=3 \end{align*}$$

\]
The angle \(\angle CDB=9x-2=9\times3-2 = 25^{\circ}\).

Answer:

Problem 10: \(\angle T = 65^{\circ}\)
Problem 11: \(\angle D = 31^{\circ}\)
Problem 12: \(\angle CAB=140^{\circ}\)
Problem 13: \(\angle C = 50^{\circ}\)
Problem 14: \(\angle CDB=25^{\circ}\)