Parallel Lines Study Guide

The big idea

Eight angles, only two sizes

When a transversal crosses two parallel lines, it makes eight angles. But there are only two different sizes: every acute angle is equal, every obtuse angle is equal, and one of each always adds up to 180°.

The four acute angles: all 60° The four obtuse angles: all 120°

Congruent equal measures

Corresponding
Alternate interior
Alternate exterior
Vertical

Supplementary add up to 180°

Same-side interior
Same-side exterior
Linear pair

The arrow marks (›) on lines ℓ and m mean they are parallel. These rules only work when the lines are parallel, except vertical angles and linear pairs, which work for any two lines that cross.

The shortcut

With parallel lines, if two angles are both acute or both obtuse, they're equal. If one is acute and one is obtuse, they add to 180°. You still need the pair's name for proofs, but this tells you which equation to write.

Lines & angles

Angle pairs

A transversal is a line that crosses two or more lines. Angles between the two lines are interior. Angles outside them are exterior. Each pair's name tells you where its two angles are.

Try it: pick a pair

Pick a pair above, or click any two angles in the drawing to see how they're related.

Tilt the transversal60° Show angle measures

Corresponding angles

Congruent

Where: the same spot at each intersection (both upper-left, both lower-right, and so on).

Look for: an F shape. Pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8.

Alternate interior angles

Congruent

Where: between the lines, on opposite sides of the transversal.

Look for: a Z shape. Pairs: ∠3 & ∠6, ∠4 & ∠5.

Alternate exterior angles

Congruent

Where: outside the lines, on opposite sides of the transversal.

Pairs: ∠1 & ∠8, ∠2 & ∠7.

Vertical angles

Always congruent

Where: across from each other at the same intersection. They make an X.

Pairs: ∠1 & ∠4, ∠2 & ∠3, ∠5 & ∠8, ∠6 & ∠7. True for any two crossing lines.

Same-side interior angles

Supplementary

Where: between the lines, on the same side of the transversal. Also called consecutive interior angles.

Look for: a C or U shape. Pairs: ∠3 & ∠5, ∠4 & ∠6.

Same-side exterior angles

Supplementary

Where: outside the lines, on the same side of the transversal. Also called consecutive exterior angles.

Pairs: ∠1 & ∠7, ∠2 & ∠8.

Linear pair

Always supplementary

Where: side by side at the same intersection, sharing a side. Together they make a straight line (180°).

Pairs: ∠1 & ∠2, ∠1 & ∠3, ∠2 & ∠4, and so on. True for any two crossing lines.

Reading the names

Interior: between the two lines
Exterior: outside the two lines
Alternate: opposite sides of the transversal
Same-side: same side of the transversal
Corresponding: same corner at each intersection

Top row: congruent. Bottom row: supplementary.

Every pair on one page

Pair	Where the angles are	With parallel lines	Reason: lines ∥ → angles	Reason: angles → lines ∥
Corresponding	Same spot at each intersection	Congruent	Corresponding Angles Postulate	Converse of the Corresponding Angles Postulate
Alternate interior	Between the lines, opposite sides	Congruent	Alternate Interior Angles Theorem	Converse of the Alternate Interior Angles Theorem
Alternate exterior	Outside the lines, opposite sides	Congruent	Alternate Exterior Angles Theorem	Converse of the Alternate Exterior Angles Theorem
Same-side interior	Between the lines, same side	Supplementary	Same-Side (Consecutive) Interior Angles Theorem	Converse of the Same-Side Interior Angles Theorem
Same-side exterior	Outside the lines, same side	Supplementary	Same-Side (Consecutive) Exterior Angles Theorem	Converse of the Same-Side Exterior Angles Theorem
Vertical	Across from each other, same intersection	Congruent (always)	Vertical Angles Theorem	Can't prove lines parallel
Linear pair	Side by side, same intersection	Supplementary (always)	Linear Pair Postulate	Can't prove lines parallel

Kinds of lines

Parallel

In the same plane and never meet. Written ℓ ∥ m.

Perpendicular

Meet at a right angle (90°). Written a ⊥ b. The small square marks the right angle.

Intersecting

Cross at exactly one point, making two pairs of vertical angles.

Skew

Not in the same plane, and never meet. Like the two highlighted edges of a box.

Perpendicular Transversal Theorem

If a line is perpendicular to one of two parallel lines, it is perpendicular to the other one too.

Transitive Property of Parallel Lines

If a ∥ b and b ∥ c, then a ∥ c.

Algebra with angles

Solving for x

Most problems give two angles as expressions like (5x + 14)°. The angle pair tells you which equation to write.

Name the pairCorresponding? Same-side interior? Find where the two angles are.
Pick the equationCongruent pair: set them equal. Supplementary pair: add them and set the sum to 180.
Solve for xGet the x terms on one side and the numbers on the other.
Answer the questionIf it asks for an angle, plug x back in. Then check: acute angles should be under 90°.

Example 1 · congruent pair

Alternate interior angles

Lines ℓ and m are parallel. Find x and the measure of each angle.

The angles are between the lines, on opposite sides of t: alternate interior, so they're congruent.
Set them equal: 3x + 17 = 5x − 9
Subtract 3x from both sides, then add 9: 26 = 2x, so x = 13.
Plug in: 3(13) + 17 = 56°. Check the other one: 5(13) − 9 = 56°. ✓

x = 13, and both angles are 56°

Example 2 · supplementary pair

Same-side interior angles

Lines ℓ and m are parallel. Find x and both angles.

Both angles are between the lines, on the same side of t: same-side interior, so they add to 180°.
Add them: (4x + 12) + (2x + 6) = 180
Combine like terms: 6x + 18 = 180, so 6x = 162 and x = 27.
Plug in: 4(27) + 12 = 120° and 2(27) + 6 = 60°. They add to 180°. ✓

x = 27, and the angles are 120° and 60°

Example 3 · find a different angle

Corresponding angles, then a linear pair

Lines ℓ and m are parallel. Find m∠1.

The two expressions are in the same spot (upper-left) at each intersection: corresponding, so they're equal.
6x − 2 = 4x + 22, so 2x = 24 and x = 12.
Those angles are 6(12) − 2 = 70°.
∠1 makes a linear pair with the 70° angle next to it, so m∠1 = 180° − 70° = 110°.

m∠1 = 110° (not 70°: always answer what was asked)

Example 4 · two crossing lines

Vertical angles

Find x. (No parallel lines needed: vertical angles are always equal.)

The angles are across from each other at the same point: vertical, so they're congruent.
7x − 4 = 5x + 12, so 2x = 16 and x = 8.
Each angle is 7(8) − 4 = 52°.

x = 8 (each angle is 52°)

Example 5 · angle bisector

A ray that splits an angle in half

BD bisects ∠ABC. Find m∠ABC.

A bisector makes two congruent angles: 3x + 4 = 5x − 14.
18 = 2x, so x = 9.
Each half is 3(9) + 4 = 31°.
The whole angle is twice that: 2 × 31° = 62°.

m∠ABC = 62°

Common mistakes

Using the wrong equation. Setting supplementary angles equal, or adding congruent angles to 180. Name the pair first.
Stopping at x. If the question asks for an angle, plug x back in.
Answering the wrong angle. If the angle asked for isn't one of the expressions, use a vertical angle or linear pair to get to it (Example 3).
Not checking. Plug x into both expressions. Congruent pairs should match. Supplementary pairs should add to 180°.

Reasoning

Two-column proofs

A proof is a chain of statements, each backed by a reason. Start with the Given. End with what you need to Prove. Every step in between needs a definition, property, postulate, or theorem.

You know the lines are parallel

Use the theorem

Lines ∥ → angles. Example: “ℓ ∥ m, so ∠3 ≅ ∠6” uses the Alternate Interior Angles Theorem.

You want to prove lines are parallel

Use the converse

Angles → lines ∥. Example: “∠3 ≅ ∠6, so ℓ ∥ m” uses the Converse of the Alternate Interior Angles Theorem.

Reasons you'll use

Properties of equality

Addition / Subtraction / Multiplication / Division: Do the same thing to both sides. If x + 9 = 20, then x = 11 (Subtraction).
Substitution: Swap in something equal. If m∠1 = 50° and m∠1 + m∠2 = 180°, then 50° + m∠2 = 180°.
Distributive: 2(x + 5) = 2x + 10
Reflexive · Symmetric · Transitive: a = a · If a = b, then b = a · If a = b and b = c, then a = c.

Properties of congruence

Reflexive: ∠A ≅ ∠A
Symmetric: If ∠A ≅ ∠B, then ∠B ≅ ∠A.
Transitive: If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. This is how you link two congruences together.

Definitions (they work both ways)

Congruent angles: ∠1 ≅ ∠2 means m∠1 = m∠2. Use it to switch between ≅ and =.
Angle bisector: A ray that splits an angle into two congruent angles.
Supplementary / Complementary: Two angles that add to 180° / to 90°.
Right angle · Perpendicular lines: A right angle measures 90°. Perpendicular lines meet at right angles.

Postulates & theorems

Angle Addition Postulate: If D is inside ∠ABC, then m∠ABD + m∠DBC = m∠ABC.
Linear Pair Postulate: A linear pair is supplementary. (Some books: Supplement Theorem.)
Vertical Angles Theorem: Vertical angles are congruent.
The parallel-line angle theorems and their converses: See the table in Angle pairs.

Proof tips

Write the Given first, one fact per line, with the reason “Given.”
Mark the diagram. Shade the angles you know are equal. The proof usually connects them in a chain.
Work backwards from the Prove. To prove lines parallel, you need a congruent (or supplementary) angle pair, so find which pair you can reach.
To combine “∠1 ≅ ∠3” and “∠1 ≅ ∠2” into “∠3 ≅ ∠2,” use the Transitive Property (or Substitution).

Worked proofs

These match the kinds of proofs in this unit. Try each step before you reveal it.

Proof 1 · proving lines are parallel

Two facts, then a converse

Givenp ∥ q, ∠1 ≅ ∠2

Prover ∥ s

#	Statements	Reasons
1	p ∥ q	Given
2	∠1 ≅ ∠3	Corresponding Angles Postulate (∠1 and ∠3 are in the same spot where r crosses p and q)
3	∠1 ≅ ∠2	Given
4	∠3 ≅ ∠2	Transitive Property of Congruence (steps 2 and 3)
5	r ∥ s	Converse of the Alternate Interior Angles Theorem (∠3 and ∠2 are alternate interior angles along q)

Why it works: the given parallel lines get you a congruent pair (step 2). Chain it to the given pair (step 4), and you have a congruent pair that proves the other two lines are parallel.

Proof 2 · three parallel lines

Linking two angle facts

Givena ∥ b, b ∥ c

Prove∠1 ≅ ∠3

#	Statements	Reasons
1	a ∥ b	Given
2	∠1 ≅ ∠2	Alternate Interior Angles Theorem (between a and b, opposite sides of t)
3	b ∥ c	Given
4	∠2 ≅ ∠3	Corresponding Angles Postulate (same spot where t crosses b and c)
5	∠1 ≅ ∠3	Transitive Property of Congruence (steps 2 and 4)

Another way: a ∥ c by the Transitive Property of Parallel Lines. Then ∠1 and ∠3 are alternate interior angles for a and c, so ∠1 ≅ ∠3.

Proof 3 · parallel segments and a bisector

Using a bisector inside a triangle

GivenDE ∥ KL; DE bisects ∠KDJ

Prove∠4 ≅ ∠5

#	Statements	Reasons
1	DE ∥ KL	Given
2	DE bisects ∠KDJ	Given
3	∠1 ≅ ∠2	Definition of angle bisector
4	∠1 ≅ ∠4	Corresponding Angles Postulate (side JL is a transversal for DE and KL)
5	∠2 ≅ ∠5	Alternate Interior Angles Theorem (transversal DK)
6	∠4 ≅ ∠5	Transitive Property of Congruence (steps 3, 4, and 5)

Why it works: the parallel segments connect ∠1 to ∠4 and ∠2 to ∠5, and the bisector connects ∠1 to ∠2. If a problem gives you the angle measures instead (like m∠4 = m∠5) and asks you to prove the bisector, use the same three connections in the other order, ending with “Definition of angle bisector.”

Some teachers want a separate line for “m∠1 = m∠2” (Definition of congruent angles) whenever you switch between ≅ and =. Ask what yours expects.

Reasoning

Conditional statements & logic

A conditional is an if-then statement. The “if” part is the hypothesis (p) and the “then” part is the conclusion (q).

If it is raining, then the ground is wet.

p = hypothesisq = conclusion→ “if… then”~ “not”↔ “if and only if”

The four forms

Name	Symbols	How to make it	Example	Same truth as the original?
Conditional	p → q	The original	If it is raining, then the ground is wet.	It is the original
Converse	q → p	Switch the two parts	If the ground is wet, then it is raining.	Not always (a sprinkler could make it wet)
Inverse	~p → ~q	Negate both parts	If it is not raining, then the ground is not wet.	Not always
Contrapositive	~q → ~p	Switch and negate	If the ground is not wet, then it is not raining.	Always
Biconditional	p ↔ q	Join with “if and only if”	It is raining if and only if the ground is wet.	Only if the conditional and converse are both true

Conditional = ContrapositiveAlways the same truth value (logically equivalent).

Converse = InverseAlways the same truth value as each other.

Biconditional

Only when it works both ways

“An angle is a right angle if and only if it measures 90°” is true: both “if right, then 90°” and “if 90°, then right” are true.

“Two angles are congruent if and only if they are vertical” is false: congruent angles don't have to be vertical.

Counterexample

One example that breaks it

To show a conditional is false, find a case where the hypothesis is true but the conclusion is false.

“If two angles are congruent, then they are vertical angles.” Counterexample: alternate interior angles with parallel lines are congruent, but not vertical.

Laws of logic (deductive reasoning)

Law of Detachment

If p → q is true and p is true, then q is true.

If a figure is a square, then it has four right angles.

ABCD is a square.

∴ ABCD has four right angles.

Law of Syllogism

If p → q and q → r, then p → r.

If it snows, then school is canceled.

If school is canceled, then I sleep in.

∴ If it snows, then I sleep in.

Invalid arguments to watch for

Starting from the conclusion

If you study, then you pass.

Jordan passed.

∴ Jordan studied.

Invalid. Jordan could have passed another way. Knowing q doesn't tell you p.

Denying the hypothesis

If you study, then you pass.

Jordan didn't study.

∴ Jordan didn't pass.

Invalid. That's using the inverse, which isn't guaranteed.

A chain that doesn't link

If Sam practices, then he makes the team.

If Ava practices, then she makes the team.

∴ If Sam practices, then Ava makes the team.

Invalid. For syllogism, the conclusion of the first statement must be the hypothesis of the second.

Inductive reasoning

Look at examples or a pattern and make a conjecture (an educated guess). It could be wrong: one counterexample disproves it.

Deductive reasoning

Use facts, definitions, and the laws of logic to reach a conclusion that must be true. Proofs use deductive reasoning.

Glossary

Key words

Transversal: A line that crosses two or more lines at different points.
Parallel lines: Lines in the same plane that never intersect (∥).
Perpendicular lines: Lines that meet at right angles (⊥).
Skew lines: Lines that aren't in the same plane and never intersect.
Interior angles: Angles between the two lines a transversal crosses.
Exterior angles: Angles outside the two lines.
Corresponding angles: Same spot at each intersection. Congruent if the lines are parallel.
Alternate interior angles: Between the lines, opposite sides of the transversal. Congruent if the lines are parallel.
Alternate exterior angles: Outside the lines, opposite sides. Congruent if the lines are parallel.
Same-side interior angles: Between the lines, same side (consecutive interior). Supplementary if the lines are parallel.
Same-side exterior angles: Outside the lines, same side. Supplementary if the lines are parallel.
Vertical angles: Opposite angles formed by two crossing lines. Always congruent.
Linear pair: Two adjacent angles that form a straight line. Always supplementary.
Congruent (≅): Exactly the same size. Congruent angles have equal measures.
Supplementary: Two angles that add up to 180°.
Complementary: Two angles that add up to 90°.
Angle bisector: A ray that divides an angle into two congruent angles.
Conditional: An if-then statement, p → q.
Hypothesis / Conclusion: The “if” part (p) / the “then” part (q).
Converse: q → p: switch the parts.
Inverse: ~p → ~q: negate both parts.
Contrapositive: ~q → ~p: switch and negate. Always as true as the original.
Biconditional: p ↔ q, “if and only if.” True only when it works both ways.
Negation: The opposite of a statement (~p, “not p”).
Counterexample: An example where the hypothesis is true but the conclusion is false.
Law of Detachment: p → q and p, so q.
Law of Syllogism: p → q and q → r, so p → r.
Postulate: A statement accepted as true without proof.
Theorem: A statement that has been proven.

Eight angles, only two sizes

Congruent equal measures

Supplementary add up to 180°

Angle pairs

Corresponding angles

Alternate interior angles

Alternate exterior angles

Vertical angles

Same-side interior angles

Same-side exterior angles

Linear pair

Reading the names

Every pair on one page

Kinds of lines

Parallel

Perpendicular

Intersecting

Skew

Solving for x

Alternate interior angles

Same-side interior angles

Corresponding angles, then a linear pair

Vertical angles

A ray that splits an angle in half

Two-column proofs

Use the theorem

Use the converse

Reasons you'll use

Properties of equality

Properties of congruence

Definitions (they work both ways)

Postulates & theorems

Worked proofs

Two facts, then a converse

Linking two angle facts

Using a bisector inside a triangle

Conditional statements & logic

The four forms

Only when it works both ways

One example that breaks it

Laws of logic (deductive reasoning)

Law of Detachment

Law of Syllogism

Invalid arguments to watch for

Starting from the conclusion

Denying the hypothesis

A chain that doesn't link

Key words

Practice test

Done