i don’t know what i’m doing!

A "transversal" is a line that cuts across two parallel lines. At each intersection, 4 angles are formed. The angles are given different names, so we can talk about pairs of them being congruent.

The four angles between the parallel lines are called interior angles. The four angles outside the parallel lines are called exterior angles. When the angles are on opposite sides of the transversal, they are alternate angles.

In the diagram, we can identify the following pairs in each category:

alternate interior: {3, 9}, {5, 10}
alternate exterior: {1, 12}, {7, 52°}

When interior angles are on the same side of the transversal, they are called same-side or consecutive interior angles. Exterior angles cannot be consecutive. Here are some in that category:

consecutive interior: {3, 6}, {5, 87°}

Angles created by a ray extending from a line are a linear pair. Angles of a linear pair are supplementary, that is, their sum is 180°. Angles formed by two intersecting lines, sharing only the same vertex, are called vertical angles. Vertical angles are both supplementary to the other angle of the linear pair of which they are a part. Since they are supplementary to the same angle, they are congruent (have the same measure). Here are some linear pairs and some vertical angles in the figure:

linear pairs: {6, 7}, {7, 8}, {8, 9}, {6, 9}, {10, 87°}, {10, 11}, {11, 12}, {12, 87°}
vertical angles: {1, 5}, {2, 4}, {3, 52°}, {6, 8}, {7, 9}, {10, 12}, {11, 87°}

Corresponding angles are ones that are in the same direction from the point of intersection. Some of those pairs are ...

corresponding angles: {1, 10}, {5, 12}, {3, 7}, {9, 52°}

Here are the relations that help you work this problem:

alternate interior angles are congruent
alternate exterior angles are congruent
vertical angles are congruent
corresponding angles are congruent
a linear pair is supplementary
consecutive interior angles are supplementary

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So far, we haven't mentioned much about the angles where lines j, k, l all meet. Transversal j cuts some of the angles created by transversal k, and vice versa. So, there are some angle sum relations that also apply to corresponding angles:

∠1+∠2≅∠6
∠2+52°≅87°
∠3+∠4≅∠11
∠4+∠5≅∠8

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With an awareness of all of the above, you can figure the measures of all of the angles in the diagram.

∠1 ≅ ∠5 ≅ ∠10 ≅ ∠12 = 180° -87° = 93°

∠2+52° = 87° ⇒ ∠2 ≅ ∠4 = 87° -52° = 35°

∠3 ≅ ∠7 ≅ ∠9 ≅ 52°

∠6 ≅ ∠8 = 180° -∠7 = 128°

∠11 ≅ 87°