01 / 21

Theorems about
Lines & Angles

Master the key theorems: vertical angles, parallel line transversals, and perpendicular bisectors—the building blocks of geometric proof.

Geometry

What Are Angle Theorems?

Angle theorems are proven statements about the relationships between angles formed when lines intersect or when a transversal crosses parallel lines. They let us deduce unknown angle measures and write rigorous proofs.

Vertical Angles
Formed by two intersecting lines, always congruent
Parallel Line Angles
Created by a transversal cutting parallel lines
Perpendicular Bisector
Points equidistant from a segment's endpoints

Vertical Angles Defined

1 2 3 4
\(\angle 1 \cong \angle 3\)
\(\angle 2 \cong \angle 4\)

Vertical Angles Are Congruent

1
Linear pair: \(\angle 1\) and \(\angle 2\) are supplementary, so \(\angle 1 + \angle 2 = 180^\circ\).
2
Another linear pair: \(\angle 2\) and \(\angle 3\) are also supplementary, so \(\angle 2 + \angle 3 = 180^\circ\).
3
Subtract: Since both equal \(180^\circ\), we get \(\angle 1 + \angle 2 = \angle 2 + \angle 3\), so \(\angle 1 = \angle 3\). ✓

Parallel Lines & Transversals

l m t
\(\angle 1\)
\(\angle 4\)
\(\angle 2\)
\(\angle 5\)
\(\parallel\)

A transversal is a line that crosses two or more lines. When it crosses parallel lines, special angle pairs are formed: alternate interior and corresponding angles.

Alternate Interior Angles Theorem

\(\alpha\)
\(\alpha\)
 Interior region
\(\cong\)

Alternate interior angles lie on opposite sides of the transversal and between the parallel lines. They are always equal when the lines are parallel.

Corresponding Angles Theorem

\(\beta\)
\(\beta\)
\(\cong\)
 Same position Same position

Corresponding angles occupy the same relative position at each intersection: both above-left, both below-right, etc. When lines are parallel, these pairs are congruent.

Alternate Interior vs. Corresponding

Alternate Interior Corresponding
Side of transversal Opposite sides Same side
Position relative to parallel lines Between the lines (interior) Same position at each intersection
When lines are parallel Congruent Congruent
Common notation "Z" or "N" pattern "F" pattern

The Perpendicular Bisector Theorem

A B M P PA PB
\(PA = PB\)

Why the Perpendicular Bisector Works

1
Given: Point \(P\) lies on the perpendicular bisector of \(\overline{AB}\) at midpoint \(M\).
2
Midpoint: \(AM = MB\) by definition of bisector.
3
Right angles: \(\angle PMA = \angle PMB = 90^\circ\) by definition of perpendicular.
4
Shared side: \(PM = PM\) (reflexive property).
5
SAS congruence: \(\triangle PMA \cong \triangle PMB\), so \(PA = PB\). ✓

How to Apply These Theorems

1
Identify the configuration: Are lines intersecting, parallel with a transversal, or is there a perpendicular bisector?
2
Name the angle pair: Classify as vertical, alternate interior, corresponding, or related to a perpendicular bisector.
3
State the theorem: Write the relevant theorem as your justification.
4
Set up the equation: Use congruence (equal measures) or the supplementary relationship to solve.
5
Verify: Check that your answer is consistent with all angle relationships in the figure.

Example 1: Vertical Angles

\((3x + 10)^\circ\)
\((5x - 20)^\circ\)

Example 2: Corresponding Angles

l m
\(72^\circ\)
\((4y + 8)^\circ\)

Example 3: Perpendicular Bisector

A B M P
\(2x + 3\)
\(5x - 12\)

Common Mistakes to Avoid

✕ Assuming Parallel
Using alternate interior or corresponding angle theorems when lines are not stated or proven to be parallel.
✕ Confusing Angle Pairs
Mixing up alternate interior (opposite sides, between) with corresponding (same side, same position).
✕ Co-Interior \(\neq\) Congruent
Co-interior (same-side interior) angles are supplementary (sum to \(180^\circ\)), not congruent.

Summary of Theorems

Theorem Condition Conclusion
Vertical Angles Two lines intersect Vertical angles ≅
Alternate Interior Transversal + parallel lines Alt. int. angles ≅
Corresponding Transversal + parallel lines Corr. angles ≅
Perp. Bisector Point on perp. bisector Equidistant from endpoints

Question 1 of 5

Two lines intersect, forming four angles. One angle measures \(130^\circ\). What is the measure of the angle vertical to it?

Question 2 of 5

A transversal crosses two parallel lines. One alternate interior angle measures \((2x + 15)^\circ\) and the other measures \((4x - 5)^\circ\). What is \(x\)?

Question 3 of 5

Point \(P\) lies on the perpendicular bisector of \(\overline{AB}\). If \(PA = 3x + 7\) and \(PB = 5x - 1\), what is \(PA\)?

Question 4 of 5

Lines \(l\) and \(m\) are cut by transversal \(t\). Corresponding angles measure \(65^\circ\) and \(65^\circ\). What can you conclude?

Question 5 of 5

Two parallel lines are cut by a transversal. If one co-interior (same-side interior) angle measures \(110^\circ\), what does the other co-interior angle measure?