https://file.aiquickdraw.com/mathbot/file/1728569361411rfy2brz9.jpg

The problem involves two intersecting lines \( \overrightarrow{AB} \) and \( \overrightarrow{CD} \), which meet at point \( E \). The angles \( \angle AEC \) and \( \angle DEB \) are given as:

- \( m\angle AEC = 5x - 20^\circ \)
- \( m\angle DEB = 2x + 10^\circ \)

Since these two angles are vertical angles (opposite angles formed by intersecting lines), they must be congruent, meaning their measures are equal. Therefore, we can set up the equation:

\[
5x - 20 = 2x + 10
\]

### Solving for \( x \):

1. Subtract \( 2x \) from both sides:
   \[
   5x - 2x - 20 = 10
   \]
   \[
   3x - 20 = 10
   \]

2. Add 20 to both sides:
   \[
   3x = 30
   \]

3. Divide by 3:
   \[
   x = 10
   \]

Thus, the value of \( x \) is **10**.

Would you like further details on this solution or have any questions?

Here are 5 related questions to explore:

1. What are vertical angles, and why are they always congruent?
2. How do you solve linear equations with variables on both sides?
3. What would change if one of the angle expressions involved a different variable?
4. How do supplementary angles differ from vertical angles in this type of problem?
5. What if the given angle measures formed a supplementary pair instead of being congruent?

**Tip:** When solving equations with variables on both sides, try to isolate the variable by performing the same operation on both sides of the equation.

In the accompanying diagram, AB and CD intersect at E. m∠AEC = 5x - 20, and m∠DEB = 2x + 10. Find the value of x.

Learn how to solve for x in a problem where two lines intersect, creating vertical angles. In this geometry problem, AB and CD intersect at point E, forming two angles: m∠AEC = 5x - 20 and m∠DEB = 2x + 10. This problem uses the Vertical Angles Theorem to find x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation