To solve this, let's set up an equation based on the information given:

1. Given Information:

   • $\angle M + \angle R = 180^\circ$
   • $\angle M = (4x + 20)^\circ$
   • $\angle R = 36^\circ$

2. Set Up Equation:
   Substitute the values of $\angle M$ and $\angle R$ into the equation for the sum of the angles: $(4x + 20) + 36 = 180$

3. Solve for $x$: $4x + 20 + 36 = 180$ Simplify: $4x + 56 = 180$ Subtract 56 from both sides: $4x = 124$ Divide by 4: $x = 31$

4. Find $\angle M$: Substitute $x = 31$ back into $\angle M = (4x + 20)$: $\angle M = 4(31) + 20 = 124 + 20 = 144^\circ$

5. Verify: $\angle M + \angle R = 144^\circ + 36^\circ = 180^\circ$

Solution: $x = 31$, and the measure of $\angle M$ is $144^\circ$.

Would you like a further breakdown, or do you have questions about any of these steps?

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1. If $x = 31$, what would be the measure of an angle adjacent to $\angle M$ in a different scenario?
2. How would the equation change if $\angle R$ were $(5x + 10)^\circ$ instead of $36^\circ$?
3. What is the general rule for finding the measure of angles that add up to $180^\circ$?
4. Could you find $x$ if the sum of $\angle M$ and $\angle R$ was not $180^\circ$, but rather $150^\circ$?
5. How can you verify whether two angles are supplementary?

Tip: In geometry, angles that add up to $180^\circ$ are known as supplementary angles.