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Angles and Parallel Lines
When a line crosses two parallel lines, eight angles appear, and they only come in two sizes.
This unit covers the names for angles and angle pairs, the rules for parallel lines, and why a triangle’s angles always add to 180°.
Angle basics
An angle is made by two rays that start at the same point. That point is the vertex, and the rays are the angle’s sides. We name an angle by its vertex, as in ∠B, or by three points with the vertex in the middle, as in ∠ABC. Angles are measured in degrees, and a full turn is 360°.
| Type | Size |
|---|---|
| Acute | more than 0° and less than 90° |
| Right | exactly 90°, drawn with a small square |
| Obtuse | more than 90° and less than 180° |
| Straight | exactly 180°, a straight line |
| Reflex | more than 180° and less than 360° |
Two angles with the same measure are congruent. Angles that share a vertex and a side but don’t overlap are adjacent.
Angle pairs
Some pairs of angles have names because their sizes are tied together.
- Complementary angles add to 90°. The complement of 35° is 55°.
- Supplementary angles add to 180°. The supplement of 35° is 145°.
- A linear pair is two adjacent angles whose outer sides form a straight line. They are always supplementary.
- Vertical angles are the angles across from each other where two lines cross. They are always equal.
Why vertical angles are equal. In the figure, ∠1 and ∠2 make a straight line, so ∠1 + ∠2 = 180°. ∠2 and ∠3 also make a straight line, so ∠2 + ∠3 = 180°.
Both sums equal 180°, so ∠1 + ∠2 = ∠2 + ∠3. Take ∠2 away from both sides and you get ∠1 = ∠3.
Euclid proves the same two facts in the Elements: linear pairs in Book I, Proposition 13, and vertical angles in Proposition 15.
A transversal across parallel lines
Parallel lines are lines in the same plane that never meet, however far you extend them. We write m ∥ n. In diagrams, matching arrowheads on the lines mean “these are parallel.” Don’t assume lines are parallel because they look it.
A transversal is a line that crosses two other lines. Where line t crosses m and n, it makes eight angles. The four between the lines (∠3, ∠4, ∠5, ∠6) are interior angles. The four outside them (∠1, ∠2, ∠7, ∠8) are exterior angles.
| Pair | Examples | If m ∥ n |
|---|---|---|
| Corresponding | ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8 | equal |
| Alternate interior | ∠3 & ∠6, ∠4 & ∠5 | equal |
| Alternate exterior | ∠1 & ∠8, ∠2 & ∠7 | equal |
| Same-side interior | ∠3 & ∠5, ∠4 & ∠6 | add to 180° |
- Corresponding angles sit in the same position at each crossing, for example both upper right.
- Alternate means on opposite sides of the transversal.
- Same-side interior angles are also called consecutive interior angles. They are the only pair in the table that is not equal: they add to 180°.
Shortcut: when m ∥ n, all the acute angles in the figure are equal and all the obtuse angles are equal. Any acute angle plus any obtuse angle makes 180°. (If the transversal is perpendicular, all eight angles are 90°.)
These rules work only when the lines are parallel. Vertical angles and linear pairs are different: they are true at any crossing.
Explorer: turn the transversal
Proving lines are parallel
The rules above also work backwards. These are the converses. If a transversal crosses two lines and any one of these is true, the lines are parallel:
- a pair of corresponding angles is equal;
- a pair of alternate interior angles is equal;
- a pair of alternate exterior angles is equal;
- a pair of same-side interior angles adds to 180°.
So to show that two lines are parallel, measure or work out one pair. If ∠3 = 70° and ∠5 = 110°, they add to 180°, so m ∥ n. If ∠5 were 108°, the sum would be 178° and the lines would not be parallel; they would meet on the side where the angles add to less than 180°.
Euclid proves the converses first, as Book I, Propositions 27 and 28. Proposition 29 then proves the rules for parallel lines. It is the first proposition in the Elements that uses his parallel postulate (see History).
You could also start from the corresponding angles rule, accept it without proof, and prove the other rules from it. The rules come out the same either way.
Why a triangle’s angles add to 180°
The parallel line rules prove one of the most-used facts in geometry: in every triangle, the three angles add to 180°.
- Start with any triangle ABC. Through C, draw the line parallel to side AB.
- Side CA is a transversal of the two parallel lines. The angle between the new line and CA is an alternate interior angle to ∠A, so it equals a.
- In the same way, CB is a transversal, so the angle between CB and the new line equals b.
- At C, the angles a, c and b fill a straight line, so a + b + c = 180°.
This is essentially Euclid’s argument in Book I, Proposition 32 (he extends side BC instead).
The exterior angle theorem
Extend one side of a triangle. The angle outside the triangle is an exterior angle. It equals the sum of the two interior angles that are not next to it (the remote interior angles).
Why: d and the triangle’s angle at B form a linear pair, so d = 180° − ∠B. The triangle sum gives a + c = 180° − ∠B too. So d = a + c.
Euclid states this in the same proposition, I.32, just before the angle sum.
Worked examples
Example 1: Fill in all eight angles
Lines m ∥ n and ∠2 = 58° (numbering as in the figure above). Find the other seven.
- ∠3 is vertical to ∠2, so ∠3 = 58°. ∠6 is corresponding to ∠2 and ∠7 is alternate exterior to ∠2, so both are 58°.
- ∠1 forms a linear pair with ∠2, so ∠1 = 180° − 58° = 122°.
- ∠4, ∠5 and ∠8 are equal to ∠1, so they are 122° too.
Example 2: Solve for x with alternate interior angles
m ∥ n, ∠3 = (3x + 10)° and ∠6 = (5x − 30)°. Find x.
- ∠3 and ∠6 are alternate interior angles, and the lines are parallel, so they are equal: 3x + 10 = 5x − 30.
- Subtract 3x and add 30: 40 = 2x, so x = 20.
- Check: 3(20) + 10 = 70 and 5(20) − 30 = 70. Both angles are 70°.
Example 3: Same-side interior angles
m ∥ n, ∠4 = (2x + 15)° and ∠6 = (3x − 10)°. Find x and both angles.
- Same-side interior angles add to 180°: (2x + 15) + (3x − 10) = 180.
- 5x + 5 = 180, so 5x = 175 and x = 35.
- The angles are 2(35) + 15 = 85° and 3(35) − 10 = 95°. Check: 85 + 95 = 180.
Tip: first name the pair, then write the equation. Equal pairs give “one angle = the other.” Same-side interior angles and linear pairs give “one angle + the other = 180.”
Example 4: Angles in a triangle
A triangle’s angles are x, 2x and 3x degrees. Find them.
- x + 2x + 3x = 180, so 6x = 180 and x = 30.
- The angles are 30°, 60° and 90°, so this is a right triangle.
Example 5: An exterior angle
An exterior angle of a triangle is (4x − 5)°. The two remote interior angles are (x + 20)° and (2x + 5)°. Find x.
- Exterior angle theorem: 4x − 5 = (x + 20) + (2x + 5), so 4x − 5 = 3x + 25.
- x = 30. The exterior angle is 115°, and the remote interior angles are 50° and 65°.
- Check: the third angle is 180° − 115° = 65°, and 50 + 65 + 65 = 180.
Common mistakes
A short history: Euclid’s fifth postulate
Euclid’s Elements, written around 300 BCE, builds geometry from a short list of starting assumptions. Four of his five postulates are short, such as “all right angles are equal to one another.” The fifth, the parallel postulate, is longer:
“That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
In this unit’s words: if same-side interior angles add to less than 180°, the lines are not parallel and meet on that side.
Euclid holds off on using it. Book I, Propositions 27 and 28 prove the converses (equal alternate angles mean the lines are parallel) without it. Proposition 29, the rules for parallel lines, is the first that needs it. Proposition 32 then gives the triangle angle sum and the exterior angle theorem.
For centuries, mathematicians tried to prove the fifth postulate from the other four, and every attempt failed. In 1795 John Playfair popularized an equivalent form, now called Playfair’s axiom: through a point not on a line, there is exactly one line parallel to it. Others found that the postulate is equivalent to the triangle angle sum being 180°.
In the 1800s, Nikolai Lobachevsky (who published in 1829) and János Bolyai, working separately, built a consistent geometry in which the fifth postulate is false. Gauss had reached similar ideas earlier but never published them. This is now called non-Euclidean geometry. In their geometry there is more than one parallel through a point, and triangle angles add to less than 180°. On the surface of a sphere, another kind of non-Euclidean geometry, there are no parallel lines at all, and triangle angles add to more than 180° (MathWorld: Spherical Triangle).
So which is right?
Each geometry is consistent on its own. Euclid’s is the one that fits a flat sheet of paper, and it is the one used in school geometry and on this site. The others describe curved surfaces and spaces.
Practice: Find the Angle
You’ll get three kinds: name an angle pair, find a missing angle, and solve for x. Type whole numbers; you can leave off the degree sign.
Sources
- Euclid, Elements, Book I, ed. D. E. Joyce, Clark University: definitions and postulates, including Postulate 5; Proposition 13 (linear pairs); Proposition 15 (vertical angles).
- Joyce’s Elements, Book I: Proposition 27 and Proposition 28 (proving lines parallel); Proposition 29 (angles made by parallel lines, with Joyce’s note that it is the first proposition to use the parallel postulate); Proposition 32 (exterior angle and triangle angle sum).
- Eric W. Weisstein, “Spherical Triangle,” MathWorld (Wolfram Research): Spherical Triangle, for the angle sum of a spherical triangle.
- J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: Non-Euclidean geometry (the fifth postulate, Playfair’s axiom, Legendre, Gauss, Bolyai, Lobachevsky and Riemann).
- MacTutor: Euclid of Alexandria (biography).
Credits: all diagrams on this page are original SVG drawings made for Jugami. The triangle sum proof is essentially Euclid’s argument in Elements I.32 (he extends side BC instead), redrawn here. Quotations from the Elements follow the translation on Joyce’s site.