Geometry worksheet focusing on angle properties at a point, featuring six problems with intersecting lines and labeled angles to solve for x.

I'll solve each of the six geometry problems by applying angle properties. Let me work through them one by one.

Problem 1

Looking at the first diagram, I see two parallel lines cut by a transversal. The angle marked 65° and angle x are corresponding angles (they're in the same relative position at each intersection). When parallel lines are cut by a transversal, corresponding angles are equal.

Therefore, x = 65°.

Problem 2

In the second diagram, I see two intersecting lines forming four angles. The angle marked 114° and angle x are vertically opposite angles (they're opposite each other at the intersection point). Vertically opposite angles are always equal.

Therefore, x = 114°.

Problem 3

For the third diagram, I see two parallel lines cut by a transversal. The angle marked 120° and angle x are alternate interior angles (they're on opposite sides of the transversal and between the parallel lines). When parallel lines are cut by a transversal, alternate interior angles are equal.

Therefore, x = 120°.

Problem 4

In the fourth diagram, I see two parallel lines cut by a transversal. The angle marked 55° and angle x are consecutive interior angles (they're on the same side of the transversal and between the parallel lines). Consecutive interior angles are supplementary, meaning they add up to 180°.

So, 55° + x = 180°
x = 180° - 55°
x = 125°

Problem 5

For the fifth diagram, I see two parallel lines cut by a transversal. The angle marked 74° and angle x are alternate exterior angles (they're on opposite sides of the transversal and outside the parallel lines). When parallel lines are cut by a transversal, alternate exterior angles are equal.

Therefore, x = 74°.

Problem 6

In the sixth diagram, I see two intersecting lines forming four angles. The angle marked 118° and angle x are adjacent angles that form a straight line (they're next to each other and together make a 180° angle). Adjacent angles on a straight line are supplementary, meaning they add up to 180°.

So, 118° + x = 180°
x = 180° - 118°
x = 62°

## Final Answers:
1. x = 65°
2. x = 114°
3. x = 120°
4. x = 125°
5. x = 74°
6. x = 62°