5. Parallel and Intersecting lines - Questions

Here are the step-by-step solutions for the angles marked a through j in the image, based on geometric principles:

1. a = 48°: The angle marked a and the 48° angle are vertically opposite angles, which are always equal.

2. b = 52°: The lines are marked as parallel with arrows. The angle b and the 52° angle are alternate interior angles, so they are equal.

4. d = 99° d° is vertically opposite to 99° (formed by intersecting lines), since 81° and 99° are linear pair (81+99=180 ✓).
   d = 99° (vertically opposite angle to 99°)

6. f = 48°: The lines are parallel, and f and 132° are interior angles on the same side of the transversal. Interior angles on the same side always add up to 180°. Therefore, f = 180° - 132° = 48°.

7. g = 58°: The vertical lines are parallel. The angle g corresponds to the 58° angle (both are in the top-left quadrant of their respective intersections), making them equal.

8. h = 75°: The horizontal lines are parallel. The right transversal intersects them to create alternate interior angles, so h is equal to 75°.

9. i = 54°: The two slanted lines with single arrows are parallel. The intersecting transversal creates angles of 70°, 54°, and 56° on a straight line. The interior angle between the right parallel line and the transversal is 54°. The angle i is the corresponding angle on the left parallel line, so it is also equal to 54°.

10. j = 83°: The two slanted lines with single arrows are parallel. The first transversal intersects the top parallel line at 27°, meaning it also makes a 27° alternate interior angle with the bottom parallel line. The second transversal intersects the bottom parallel line at 124°, making its adjacent interior angle 180° - 124° = 56°. These two transversals form a triangle with the bottom parallel line. The third angle of this triangle is 180° - (27° + 56°) = 97°. The angle j is adjacent to this 97° intersection on a straight line (a linear pair, which adds up to 180°). Thus, j = 180° - 97° = 83°.