Let's analyze the image step by step:

Given:

• Two lines intersect at point M.
• The angles formed at the intersection are labeled.
  • $\angle AMC = (13x - 7)^\circ$
  • $\angle CMB = (7x + 7)^\circ$

These angles, $\angle AMC$ and $\angle CMB$, are supplementary because they are adjacent angles on a straight line. The sum of supplementary angles is 180°.

Solution:

a. Find $x$:

$(13x - 7) + (7x + 7) = 180$ Simplifying: $13x - 7 + 7x + 7 = 180$ $20x = 180$ $x = 9$

So, $x = 9$.

b. Find $m\angle AMC$:

Substitute $x = 9$ into the expression for $\angle AMC$: $m\angle AMC = 13x - 7 = 13(9) - 7 = 117 - 7 = 110^\circ$

c. Find $m\angle AMD$:

Since $\angle AMC$ and $\angle AMD$ are vertically opposite angles, they are equal: $m\angle AMD = m\angle AMC = 110^\circ$

d. Find $m\angle DMB$:

Since $\angle CMB$ and $\angle DMB$ are vertically opposite angles, they are equal. We already know: $m\angle CMB = 7x + 7 = 7(9) + 7 = 63 + 7 = 70^\circ$ Thus, $m\angle DMB = 70^\circ$.

e. Find $m\angle CMB$:

We've already calculated this: $m\angle CMB = 70^\circ$

Summary of Answers:

• a. $x = 9$
• b. $m\angle AMC = 110^\circ$
• c. $m\angle AMD = 110^\circ$
• d. $m\angle DMB = 70^\circ$
• e. $m\angle CMB = 70^\circ$

Would you like more details on any step?

Here are 5 related questions for practice:

1. What is the sum of all angles formed at the intersection point M?
2. If $\angle AMC$ was reduced by 10°, how would it affect $x$?
3. How do vertically opposite angles help in solving for unknowns in similar problems?
4. If the line AB was parallel to line CD, how would that affect the angle relationships?
5. Could the angles be complementary in any setup like this?

Tip: Always look for angle pairs (like vertical, supplementary) when two lines intersect; they simplify calculations!