Question

1. if m∠7 = -2x - 1 and m∠2 = 9x + 46, solve for x and find m∠7 and m∠2.

Explanation:

Step1: Assume ∠7 and ∠2 are equal (vertical or alternate interior angles, typical in geometry for parallel lines and transversals)

Set \( -2x - 1 = 9x + 46 \)

Step2: Solve for x

Add \( 2x \) to both sides: \( -1 = 11x + 46 \)
Subtract 46: \( -47 = 11x \) → \( x = -\frac{47}{11} \) (Wait, maybe they are supplementary? Let's check again. If ∠7 and ∠2 are supplementary (linear pair or same - side interior), then \( (-2x - 1)+(9x + 46)=180 \))

Step3: Correct equation (supplementary angles, sum to 180°)

\( -2x - 1+9x + 46 = 180 \)
Combine like terms: \( 7x + 45 = 180 \)
Subtract 45: \( 7x = 135 \) → \( x=\frac{135}{7}\approx19.29 \) (Wait, maybe the original problem has a typo, or maybe ∠7 and ∠2 are equal. Let's re - examine the problem. If we assume they are equal (vertical angles), then:
\( -2x - 1=9x + 46 \)
\( -2x-9x = 46 + 1 \)
\( -11x = 47 \)
\( x=-\frac{47}{11}\approx - 4.27 \)
Then \( m\angle7=-2\times(-\frac{47}{11})-1=\frac{94}{11}-1=\frac{94 - 11}{11}=\frac{83}{11}\approx7.55 \)
\( m\angle2=9\times(-\frac{47}{11})+46=-\frac{423}{11}+\frac{506}{11}=\frac{83}{11}\approx7.55 \)

But if we assume supplementary (sum to 180°):
\( 7x+45 = 180 \)
\( 7x=135 \)
\( x = \frac{135}{7}\approx19.29 \)
\( m\angle7=-2\times\frac{135}{7}-1=-\frac{270}{7}-1=-\frac{277}{7}\approx - 39.57 \) (invalid, angle measure can't be negative)

So the correct assumption is that ∠7 and ∠2 are equal (vertical angles or alternate interior angles, so non - negative measures). So we proceed with \( -2x - 1=9x + 46 \)

Step1 (revised): Set equal angles

\( -2x-1 = 9x + 46 \)

Step2 (revised): Solve for x

\( -2x-9x=46 + 1 \)
\( -11x = 47 \)
\( x=-\frac{47}{11}\)

Step3: Find \( m\angle7 \) and \( m\angle2 \)

\( m\angle7=-2\times(-\frac{47}{11})-1=\frac{94}{11}- \frac{11}{11}=\frac{83}{11}\)
\( m\angle2=9\times(-\frac{47}{11})+46=-\frac{423}{11}+\frac{506}{11}=\frac{83}{11}\)

Answer:

\( x = -\frac{47}{11}\), \( m\angle7=\frac{83}{11}\), \( m\angle2=\frac{83}{11}\) (or in decimal form \( x\approx - 4.27\), \( m\angle7\approx7.55\), \( m\angle2\approx7.55\))