Question

1. the lines $overline{qr}$ and $overline{st}$ intersect at w such that the measure of $angle twr$ is given by $6(x - 3)$ and the measure of $angle qws$ is given by the expression $4(x + 1)$. find the value of x.

reasoning

1. could an angle be complementary to itself? if so, what would its measure be?
2. could an angle be supplementary to itself? if so, what would its measure be?

Explanation:

Step1: Identify vertical - angles property

Vertical angles are equal. $\angle TWR$ and $\angle QWS$ are vertical angles, so $6(x - 3)=4(x + 1)$.

Step2: Expand both sides

Expand the left - hand side: $6x-18$, and the right - hand side: $4x + 4$. So, $6x-18=4x + 4$.

Step3: Isolate the variable terms

Subtract $4x$ from both sides: $6x-4x-18=4x-4x + 4$, which simplifies to $2x-18 = 4$.

Step4: Isolate the variable

Add 18 to both sides: $2x-18 + 18=4 + 18$, getting $2x=22$.

Step5: Solve for x

Divide both sides by 2: $\frac{2x}{2}=\frac{22}{2}$, so $x = 11$.

Step6: Answer question 8

Let the angle be $x$. Complementary angles add up to $90^{\circ}$. If an angle is complementary to itself, then $x+x=90^{\circ}$, or $2x = 90^{\circ}$. Solving for $x$, we get $x = 45^{\circ}$. So an angle can be complementary to itself, and its measure is $45^{\circ}$.

Step7: Answer question 9

Let the angle be $y$. Supplementary angles add up to $180^{\circ}$. If an angle is supplementary to itself, then $y + y=180^{\circ}$, or $2y=180^{\circ}$. Solving for $y$, we get $y = 90^{\circ}$. So an angle can be supplementary to itself, and its measure is $90^{\circ}$.

Answer:

For question 7: $x = 11$
For question 8: Yes, $45^{\circ}$
For question 9: Yes, $90^{\circ}$