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

To solve the problem, let's denote the two equal angles, Angle 1 and Angle 2, as \( x \). Given the information, we have:

- Angle 1 = \( x \)
- Angle 2 = \( x \)
- Angle 3 = \( 170^\circ \)

Now, Angle 4 can measure either \( 170^\circ - 55^\circ = 115^\circ \) or \( 170^\circ + 55^\circ = 225^\circ \).

Since a quadrilateral's angles must sum to \( 360^\circ \), we can create equations for both cases of Angle 4.

### Case 1: Angle 4 = 115 degrees
Using the equation for the sum of the angles in a quadrilateral, we have:

\[
x + x + 170^\circ + 115^\circ = 360^\circ
\]

Combining like terms, we get:

\[
2x + 285^\circ = 360^\circ
\]

Subtract \( 285^\circ \) from both sides:

\[
2x = 360^\circ - 285^\circ
\]
\[
2x = 75^\circ
\]

Now divide by 2:

\[
x = \frac{75^\circ}{2} = 37.5^\circ
\]

Thus, in this case, Angle 2 (which equals Angle 1) is:

\[
\text{Angle 2} = x = 37.5^\circ
\]

### Case 2: Angle 4 = 225 degrees
Using the equation for the sum of angles again:

\[
x + x + 170^\circ + 225^\circ = 360^\circ
\]

Combining like terms, we have:

\[
2x + 395^\circ = 360^\circ
\]

Subtract \( 395^\circ \) from both sides:

\[
2x = 360^\circ - 395^\circ
\]
\[
2x = -35^\circ
\]

Dividing by 2 gives us:

\[
x = -17.5^\circ
\]

Since angle measures cannot be negative, this case is not valid.

### Conclusion
The only valid solution occurs in Case 1, where:

\[
\text{Angle 2} = 37.5^\circ
\]

Thus, the angle measurement of Angle 2 is:

\[
\boxed{37.5^\circ}
\]