Answer:
Without knowing the specific diagram, it is difficult to give a step-by-step proof. However, if lines k and m intersect at point P, we can use the following reasoning:
- The angles formed by intersecting lines are either congruent or supplementary.
- Angles 1 and 3 are opposite each other, meaning they are vertical angles. By definition, vertical angles are congruent.
- Angles 2 and 3 are alternate interior angles, meaning they are on opposite sides of the transversal line and between the two intersected lines. When two lines are cut by a transversal and alternate interior angles are congruent.
- Therefore, angles 1 and 3 are congruent because they are vertical angles, and angles 2 and 4 are congruent because they are alternate interior angles.
Alternatively, we could use the following proof:
- Draw a line n that passes through point P and is parallel to line k.
- Since line n is parallel to line k, angle 1 and angle 2 are corresponding angles and are therefore congruent.
- Draw a line l that passes through point P and is parallel to line m.
- Since line l is parallel to line m, angle 3 and angle 4 are corresponding angles and are therefore congruent.
- Therefore, angle 1 is congruent to angle 2, and angle 3 is congruent to angle 4.