Question

1. name the vertex of each angle.

a) ∠5
b) ∠3

1. name the sides of each angle.

a) ∠6
b) ∠2

1. write another name for each angle.

a) ∠qpr
b) ∠1

1. a) draw a diagram in which x is the mid - point of $overline{ab}$.

b) if $overline{ax}=3x - 2$ and $overline{xb}=6x - 14$, find the length of $overline{ab}$.

1. if $mangle rts = 8x + 18$, find the value of x so that $overline{tr}perpoverline{ts}$

Explanation:

Step1: Identify vertex of angle

The vertex is the common endpoint of the rays that form the angle. For $\angle5$, looking at the diagram, the vertex is $M$. For $\angle3$, the vertex is $P$.

Step2: Identify sides of angle

The sides of an angle are the two rays that form the angle. For $\angle6$, the sides are $\overrightarrow{MN}$ and $\overrightarrow{MO}$. For $\angle2$, the sides are $\overrightarrow{PR}$ and $\overrightarrow{PQ}$.

Step3: Find alternative angle names

For $\angle QPR$, another name is $\angle1$. For $\angle1$, another name is $\angle QPR$.

Step4: Draw mid - point diagram

Draw a line segment $\overline{AB}$ and mark a point $X$ in the middle such that $AX = XB$.

Step5: Solve for length of $\overline{AB}$

Since $X$ is the mid - point of $\overline{AB}$, $AX=XB$. So, $3x - 2=6x - 14$.
Subtract $3x$ from both sides: $- 2 = 3x-14$.
Add 14 to both sides: $12 = 3x$.
Divide both sides by 3: $x = 4$.
$AX=3x - 2=3\times4 - 2=10$.
$AB=2\times AX = 20$.

Step6: Solve for $x$ when $\overline{TR}\perp\overline{TS}$

If $\overline{TR}\perp\overline{TS}$, then $m\angle RTS = 90^{\circ}$.
Set $8x + 18=90$.
Subtract 18 from both sides: $8x=72$.
Divide both sides by 8: $x = 9$.

Answer:

1. a) $M$; b) $P$
2. a) $\overrightarrow{MN}$ and $\overrightarrow{MO}$; b) $\overrightarrow{PR}$ and $\overrightarrow{PQ}$
3. a) $\angle1$; b) $\angle QPR$
4. a) (Diagram: Draw line segment $\overline{AB}$ and mark mid - point $X$); b) $20$
5. $9$