Since line j ⊥ line k, all the angles formed by the intersection of two lines is a right angle. Therefore,
∠9, ∠10, ∠11, ∠12 = 90°
By linear pair theorem, the two angles are supplementary (sum of 180°). ∠1, and ∠2 are linear pair. If ∠1 = 42°, then ∠2 = 138°.
By vertical angle theorem, the two angle are congruent, ∠2 and ∠3 are vertical angles, as well as ∠1 and ∠4, which means ∠3 = 138°, and ∠4 = 42°.
The sum of the interior angle of a triangle is equal to 180°. ∠4, ∠7, and ∠9 form an interior angle inside a triangle. Solve for ∠7 and we get
∠4 + ∠7 + ∠9 = 180°
42° + ∠7 + 90° = 180°
∠7 + 132° = 180°
∠7 = 180° - 132°
∠7 = 48°.
∠5 and ∠7 form a linear pair, they are supplementary which means ∠5 = 132°.
∠5 and ∠8 are vertical angle pairs, so is ∠7 and ∠6, this means that
∠8 = 132°, and ∠6 = 48°.
∠5 and ∠13 are corresponding angles in a two parallel lines (line l and line m) cut by a transversal (line k). Corresponding angles are congruent which means that
∠13 = 132°.
∠13 and ∠14 form a linear pair, which means ∠14 = 48°.
∠13 and ∠16 are vertical angles, so is ∠14 and ∠15.
∠15 = 48°, and ∠16 = 132°.
The Alternate Exterior Angles Theorem states that, when two parallel lines are cut by a transversal , the resulting alternate exterior angles are congruent.
∠1 and ∠20 are alternate exterior angles, which means that ∠20 = 42°.
The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal , the resulting alternate interior angles are congruent.
∠4 and ∠17 are alternate interior angles, which means ∠17 = 42°.
∠17 and ∠18 form a linear pair, ∠18 = 138°.
Lastly, ∠18 and ∠19 are vertical angles, and are therefore congruent ∠19 = 138°.