https://file.aiquickdraw.com/mathbot/file/1729199285730oejb9j5m.jpg

Let's solve part **b** first. The problem involves **supplementary angles**, which means the sum of the two angles is \(180^\circ\).

The given angles are:
\[
m\angle ABC = 20x + 14 \quad \text{and} \quad m\angle DEF = 14x - 4
\]
Since these two angles are supplementary:
\[
m\angle ABC + m\angle DEF = 180^\circ
\]
Now substitute the expressions for \(m\angle ABC\) and \(m\angle DEF\):
\[
(20x + 14) + (14x - 4) = 180
\]
Simplify the equation:
\[
20x + 14x + 14 - 4 = 180
\]
\[
34x + 10 = 180
\]
Now solve for \(x\):
\[
34x = 180 - 10
\]
\[
34x = 170
\]
\[
x = \frac{170}{34} = 5
\]

Now, for part **c**, we will substitute \(x = 5\) into the expression for \(m\angle ABC\) to find its value:
\[
m\angle ABC = 20x + 14 = 20(5) + 14 = 100 + 14 = 114^\circ
\]

### Summary:
- The value of \(x\) is \(5\).
- The measure of \(m\angle ABC\) is \(114^\circ\).

Let me know if you'd like further explanation or if you have any questions!

---

Here are five questions to explore further:
1. What is the value of \(m\angle DEF\) after solving for \(x\)?
2. How are supplementary angles different from complementary angles?
3. Can two obtuse angles be supplementary? Why or why not?
4. What is the geometric significance of supplementary angles in polygons?
5. How would this problem change if the angles were complementary instead of supplementary?

**Tip**: Always check your final angle measures to ensure they add up correctly, especially when working with supplementary or complementary angles!

If m∠ABC = 20x + 14 and m∠DEF = 14x - 4, use what you know about supplementary angles to create an equation and solve for the value of x. Show all of your work. Now, use the value of x to find m∠ABC.

Step-by-step guide to solving for x in the equation involving supplementary angles: m∠ABC = 20x + 14 and m∠DEF = 14x - 4. Includes finding the value of x and determining the measure of m∠ABC. Suitable for students in Grades 8-10.

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