Question

Leopold draws a quadrilateral with two equal angles, Angles 1 and 2. Angle 3 measures 170 degrees, and Angle 4 measures either 55 degrees less than Angle 3 or 55 degrees more than Angle 3. Determine the angle measurement of Angle 2. Show all work.(4 points)

Answer 1

Given the information in the problem, we can start by setting up our equations based on the angles of the quadrilateral.

1. **Identify the angles**:
- Let Angles 1 and 2 both be represented by \( x \) since they are equal.
- Let Angle 3 = 170 degrees.
- Let Angle 4 be either \( 170 - 55 = 115 \) degrees or \( 170 + 55 = 225 \) degrees.

2. **Quadrilateral Sum**:
In any quadrilateral, the sum of the interior angles equals \( 360 \) degrees. Therefore, we can write the equation as follows:

\[
x + x + 170 + \text{Angle 4} = 360
\]

This simplifies to:

\[
2x + 170 + \text{Angle 4} = 360
\]

Which further simplifies to:

\[
2x + \text{Angle 4} = 360 - 170
\]

\[
2x + \text{Angle 4} = 190
\]

3. **Case Analysis for Angle 4**:
Now we solve for \( x \) in each case for Angle 4.

**Case 1: Angle 4 = 115 degrees**:
\[
2x + 115 = 190
\]

Subtracting 115 from both sides gives:
\[
2x = 190 - 115
\]

\[
2x = 75
\]

Dividing by 2 gives:
\[
x = \frac{75}{2} = 37.5 \text{ degrees}
\]

So, if Angle 4 is 115 degrees, then both Angle 1 and Angle 2 measure \( 37.5 \) degrees.

**Case 2: Angle 4 = 225 degrees**:
\[
2x + 225 = 190
\]

Subtracting 225 from both sides gives:
\[
2x = 190 - 225
\]

\[
2x = -35
\]

Dividing by 2 gives:
\[
x = -\frac{35}{2} = -17.5 \text{ degrees}
\]

This case is not valid since angles cannot be negative.

4. **Conclusion**:
The only valid case is when Angle 4 is 115 degrees, which leads to Angle 1 and Angle 2 both measuring \( 37.5 \) degrees.

Therefore, the measurement of Angle 2 is:
\[
\boxed{37.5 \text{ degrees}}
\]