Angles and lines
We need to recall some properties of angles defined by crossing lines:
* Vertical or opposite angles have the same measure. Examples of vertical angles are these: 40° and angle 2, angle 1 and angle 3, angles 5 and 7, angles 4 and 6, angles 16 and 18, angles 15 and 17, angles 12 and 14, angles 13 and 120°, angles 9 and 11 and angles 8 and 10.
* Corresponding angles have the same measure. Based on the diagram, the following pairs of angles are corresponding: 40° and angle 11, angles 2 and 9, angles 1 and 8, angles 6 and 130°, angles 7 and 12, among others.
* Linear pair of angles add up to 180°. Examples of linear pairs are angles 3 and 40°, angles 1 and 2, angles 4 and 5, angles 6 and 7, among others
* Sum of angles in a triangle is 180°. Thus, the sum of the measures of angles 2, 7, and 17 is 180°. Also the sum of angles 11, 14, and 15 is 180°.
Having all of that in mind, we can find the required angles as follows:
Being 40° and angle 2 vertical, then measure of 2 = 40°
Being 40° and angle 1 linear, then measure of 1 = 180° - 40° = 140°. Measure of 1 = 140°
Angles 3 and 1 are vertical, thus measure of 3 is 140°.
Angles of 40° and 11 are corresponding, thus measure of 11 = 40°.
Angles 2 and 9 are corresponding, thus measure of 9 = 40°
Angles 3 and 10 are corresponding, measure of 10 is 140°.
Angles 1 and 8 are corresponding, thus measure of 8 = 140°
Angles 13 and 130° are vertical, thus measure of 13 is 130°.
Angles 13 and 14 are linear, thus measure of 14 is 50°.
Angles 12 and 14 are vertical, thus measure of 12 is 50°.
Angles 4 and 13 are corresponding, thus measure of 4 is 50°.
Angles 6 and 130° are corresponding, thus measure of 6 is 130°.
Angles 5 and 14 are corresponding, thus measure of 5 is 50°.
Angles 7 and 12 are corresponding, thus measure of 7 is 50°.
Angles 2, 7, and 17 sum 180°, thus angle 17 measures 180° - 40° - 50° = 90°. Measure of 17 is 90°.
Angles 15 and 17 are vertical, thus measure of 15 is 90°.
Angles 17 and 18 are linear, thus measure of 18 is 90°.
Angles 16 and 18 are vertical, thus measure of 16 is 90°.