1. In the given figure, what is the measure of angle 1?

Short Explanation:

Step-by-step solution:

1. Step 1: Identify the given angles. Angle 3 is given as 58 degrees. Angle 4 is given as 122 degrees.
2. Step 2: Recognize that angles 3 and 1 are vertically opposite angles. Vertically opposite angles are equal. Therefore, the measure of angle 1 is equal to the measure of angle 3.
3. Step 3: Calculate the measure of angle 1. Since angle 3 = 58 degrees, angle 1 = 58 degrees.
4. Step 4: Verify the solution using the straight line property. Angles 1, 2, and 3 form a straight line (or angles 4 and 3 form a straight line). Angles on a straight line sum up to 180 degrees. If angle 1 = 58 degrees and angle 3 = 58 degrees, then angle 2 = 180 - (58 + 58) = 180 - 116 = 64 degrees. Alternatively, angles 3 and 4 are adjacent angles on a straight line, so 58 + 122 = 180 degrees, which is consistent. Angles 1 and 2 are adjacent angles on a straight line. If angle 1 is 58 degrees, then angle 2 would be 180 - 58 = 122 degrees. This contradicts the initial assumption that angles 1 and 3 are vertically opposite if we consider angles 2 and 4. Let's re-examine the figure. It seems there are two intersecting lines. Let's assume the angles are labeled consecutively around the intersection point. If angle 3 is 58 degrees, then the angle vertically opposite to it is also 58 degrees. If the angles are labeled 1, 2, 3, 4 in sequence around the intersection, and angle 3 is 58 degrees, then angle 1 is vertically opposite to angle 3.

Answer: 58 degrees