Parallel Lines and Transversals

Angle relationships formed by parallel lines

Parallel Lines and Transversals

Definition

A transversal is a line that intersects two or more lines.

When a transversal crosses parallel lines, special angle relationships form.

Angle Pairs

Corresponding Angles: Same position at each intersection

Property: Congruent when lines are parallel

Alternate Interior Angles: Between the parallel lines, opposite sides

Property: Congruent when lines are parallel

Alternate Exterior Angles: Outside the parallel lines, opposite sides

Property: Congruent when lines are parallel

Consecutive Interior Angles (Same-Side Interior): Between parallel lines, same side

Property: Supplementary when lines are parallel (sum to $180°$ )

Key Theorem

If two parallel lines are cut by a transversal:

Corresponding angles are ≅
Alternate interior angles are ≅
Alternate exterior angles are ≅
Consecutive interior angles are supplementary

Converse

If these angle relationships hold, then the lines are parallel.

📚 Practice Problems

1Problem 1easy

❓ Question:

Two parallel lines are cut by a transversal. If one angle measures $65°$ , what is the measure of its corresponding angle?

💡 Show Solution

Corresponding angles are congruent when lines are parallel.

Answer: $65°$

2Problem 2medium

❓ Question:

Parallel lines $l$ and $m$ are cut by a transversal. Two consecutive interior angles measure $(2x + 10)°$ and $(3x - 15)°$ . Find $x$ .

💡 Show Solution

Consecutive interior angles are supplementary.

$(2x + 10) + (3x - 15) = 180$

$5x - 5 = 180$

$5x = 185$

$x = 37$

Answer: $x = 37$

3Problem 3hard

❓ Question:

Lines $a$ and $b$ are cut by transversal $t$ . Alternate interior angles measure $(5x - 20)°$ and $(3x + 40)°$ . Are lines $a$ and $b$ parallel?

💡 Show Solution

For the lines to be parallel, alternate interior angles must be congruent.

Set them equal: $5x - 20 = 3x + 40$

$2x = 60$

$x = 30$

When $x = 30$ :

First angle: $5(30) - 20 = 130°$
Second angle: $3(30) + 40 = 130°$

Since the angles are equal, the lines are parallel.

Answer: Yes, the lines are parallel

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