Mathematics KnowledgeLesson 14 of 21
Angles and Lines
Angle pairs, parallel lines cut by a transversal, and angle sums.
Table of ContentsShow
Angle questions are almost never hard arithmetic. They test whether you can look at a figure and see which angles are forced to be equal.
The angle types
| Type | Measure |
|---|---|
| acute | less than 90 degrees |
| right | exactly 90 degrees |
| obtuse | between 90 and 180 degrees |
| straight | exactly 180 degrees |
| reflex | between 180 and 360 degrees |
Complementary and supplementary
Complementary angles add to 90 degrees. The complement of 35 is 55.
Supplementary angles add to 180 degrees. The supplement of 35 is 145.
The pair does not have to be touching. Any two angles whose measures add to 90 are complementary, wherever they are drawn.
Two facts that follow and get asked directly:
- Angles on a straight line are supplementary. If a ray leaves a line, the two angles it makes add to 180.
- Angles around a point add to 360 degrees.
A linear pair is two angles side by side that form a straight line, so they are supplementary. If one is three times the other, the pair is x + 3x = 180, so the smaller is 45 degrees.
An angle bisector is a ray that splits an angle into two equal parts. Angles that are equal are also called congruent.
Vertical angles
When two lines cross, they make four angles. The pairs opposite each other are equal, and they are called vertical angles regardless of orientation.
This is true for any two crossing lines and needs no parallels. If one angle at a crossing is 62 degrees, the one opposite is 62, and the other two are each 118, because each of them is on a straight line with a 62.
One angle at a crossing gives you all four.
Parallel lines and a transversal
This is the figure the subtest actually draws.
Look at what the figure is telling you. There are eight angles and only two sizes. Every one of the eight is either equal to angle 1 or supplementary to it.
The named relationships:
| Name | Which | Relationship |
|---|---|---|
| Vertical | 1 and 4, 2 and 3, 5 and 8, 6 and 7 | equal |
| Corresponding | 1 and 5, 2 and 6, 3 and 7, 4 and 8 | equal |
| Alternate interior | 3 and 6, 4 and 5 | equal |
| Alternate exterior | 1 and 8, 2 and 7 | equal |
| Same-side interior | 3 and 5, 4 and 6 | sum to 180 |
Learning the names is optional. Learning that there are only two sizes is not. If a question gives you one angle and asks for another, decide whether the one you want looks like the one you were given - same orientation, same tilt - or looks like its supplement. If it looks the same, it is equal. If it looks different, subtract from 180.
That visual test is faster than recalling which pair is called what, and it does not fail under pressure.
All of this requires the lines to be parallel. If a figure does not mark them parallel with arrowheads and the question does not say they are, corresponding angles are not equal and the only thing you can still use is vertical angles.
Angles in polygons
Triangle: 180 degrees. Any triangle, always.
That single fact solves most triangle-angle questions. Given two angles, the third is 180 minus their sum.
Quadrilateral: 360 degrees.
Any polygon: (n - 2) x 180, where n is the number of sides. A pentagon has 5 sides, so (5 - 2) x 180 is 540 degrees.
The formula works because a polygon splits into n - 2 triangles from any one vertex, and each carries 180.
A regular polygon has equal angles, so divide by n. Each angle of a regular hexagon is (6 - 2) x 180 / 6, which is 720 / 6, or 120 degrees.
The exterior angle shortcut
The exterior angles of any polygon sum to 360 degrees, however many sides it has. So each exterior angle of a regular polygon is 360 / n, and the interior angle is 180 minus that.
For a regular hexagon: 360 / 6 is 60, and 180 minus 60 is 120. Same answer, less arithmetic. Reach for this whenever n is large.
What you can skip
Across the 48 questions on this topic:
- Radians and trigonometry never appear. Every angle is in degrees and found by adding or subtracting from 90, 180 or 360.
- Formal proofs. No question asks you to justify a step. The angle-pair names in this lesson are the vocabulary the questions use.
Where people lose points
Swapping complementary and supplementary. Complementary is the smaller one, 90.
Assuming parallel when the figure does not say so.
Using n instead of (n - 2) in the polygon sum.
Forgetting to divide by n when a question asks for one angle of a regular polygon rather than the total.
Reading a figure as drawn to scale. Test figures frequently are not. Use the stated measures, not what an angle looks like.
Work one in under a minute
Two parallel lines are cut by a transversal. One angle measures 115 degrees. What is the measure of its same-side interior partner?
Same-side interior angles are supplementary, so 180 minus 115 is 65 degrees.
Sanity check against the two-sizes rule: the figure contains only 115s and 65s, and 115 plus 65 is 180. Any answer that is not one of those two numbers is wrong before you check anything else.
Where this leads
The triangle sum is the basis of every triangle question, and the polygon formula carries straight into the quadrilaterals lesson.
Related lessonsReference
- Triangles and the Pythagorean Theorem - where the 180-degree sum does its work
- Quadrilaterals and Polygons - the shapes these angle sums describe
- Circles - angles measured at a center rather than between lines
- Coordinate Geometry and Slope - parallel and perpendicular expressed as slopes
Practice this topic
Check that this lesson stuck. Answer questions on angles and lines only, and see the right answer and why after each one.
Practice Angles and Lines questions