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 wo

Answer 1

Given the conditions stated in the problem, we can represent the angles of the quadrilateral as follows:

- Let \( \angle 1 = \angle 2 = x \) (since Angles 1 and 2 are equal).
- Let \( \angle 3 = 170^\circ \).
- Let \( \angle 4 \) be \( 170^\circ - 55^\circ = 115^\circ \) or \( 170^\circ + 55^\circ = 225^\circ \).

In a quadrilateral, the sum of the angles equals \( 360^\circ \). Therefore, we can set up the following equations for both cases of Angle 4 and solve for \( x \).

### Case 1: Angle 4 = 115 degrees
\[
x + x + 170^\circ + 115^\circ = 360^\circ
\]
\[
2x + 285^\circ = 360^\circ
\]
\[
2x = 360^\circ - 285^\circ
\]
\[
2x = 75^\circ
\]
\[
x = 37.5^\circ
\]

### Case 2: Angle 4 = 225 degrees
\[
x + x + 170^\circ + 225^\circ = 360^\circ
\]
\[
2x + 395^\circ = 360^\circ
\]
\[
2x = 360^\circ - 395^\circ
\]
\[
2x = -35^\circ
\]
Since angles cannot be negative, this case is invalid.

Now let's summarize the results:

From **Case 1**, we found that \( x = 37.5^\circ \).

Therefore, \( \angle 1 = 37.5^\circ \) and \( \angle 2 = 37.5^\circ \).

Thus, the measurement of Angle 2 is \( \boxed{37.5^\circ} \).