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USA - Geometry Lines and angles

Parallel lines cut by a transversal (alternate interior, corresponding, co-interior)

20 practice questions 2 video lessons Theory + worked examples

Parallel Lines and Transversals

California Geometry • Standard G-CO.9 • Lines & Angles

Parallel Lines and Transversals is a topic in Lines & Angles in the California Common Core State Standards. It is aligned to Standard G-CO.9, which requires students to prove theorems about the angles formed when parallel lines are cut by a transversal.

When parallel lines are cut by a transversal, corresponding and alternate interior angles are equal, and co-interior angles are supplementary.

California Geometry › Lines & Angles › Parallel Lines and Transversals  —  Standard G-CO.9

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Practice questions

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  • Corresponding, Alternate Interior, Alternate Exterior, Consecutive Interior Angles Watch
  • Alternate Angles, Corresponding Angles, Co-interior Angles - Nerdstudy Watch
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Theory

A transversal is a line that crosses two other lines, forming eight angles. When the two lines are parallel, the angle pairs are related:

  • Corresponding angles (same position at each crossing) — congruent.
  • Alternate interior angles (between the lines, opposite sides) — congruent.
  • Alternate exterior angles (outside the lines, opposite sides) — congruent.
  • Co-interior (same-side interior) angles — supplementary (sum \(180^\circ\)).
These relationships hold only when the lines are parallel — and, conversely, they can be used to prove lines parallel.
Parallel lines cut by a transversal A transversal crossing two parallel lines forms eight angles with equal corresponding and alternate pairs. 1 2 3 4 5 6 7 8
A transversal forms eight angles across two parallel lines.
Angle pairs (parallel lines) Angle pairs (parallel lines) Angle pairs (parallel lines) corresponding: equal alternate interior / exterior: equal co-interior (same-side): sum 180°
Which pairs are equal and which are supplementary.

The parallel-line angle rules:

\[\text{corresponding}=\text{equal},\quad \text{alternate}=\text{equal},\quad \text{co-interior}=\text{supplementary}\]
corresponding and alternate angles are equal; co-interior angles are supplementary
Converse: if one of these angle relationships holds, the lines are parallel.

How to find an angle

  1. Locate the transversal and the two parallel lines.
  2. Classify the angle pair (corresponding, alternate, co-interior).
  3. Apply the rule: equal, or supplementary.
  4. Set up and solve an equation when the angles are algebraic.
Example 1 — Corresponding angles
A transversal crosses two parallel lines. One angle is \(70^\circ\). Find its corresponding angle.
Solution

Corresponding angles are congruent.

\(\text{corresponding}\)\(=\)\(70^\circ\)
the corresponding angle is 70 degrees
Example 2 — Alternate interior angles
If \(\angle 3=110^\circ\), find its alternate interior angle \(\angle 6\).
Solution

Alternate interior angles are congruent when the lines are parallel.

\(\angle 6\)\(=\)\(110^\circ\)
the alternate interior angle is 110 degrees
Example 3 — Co-interior angles
Two co-interior (same-side interior) angles are \(x^\circ\) and \(120^\circ\). Find \(x\).
Solution

Co-interior angles are supplementary (sum to \(180^\circ\)).

\(x+120\)\(=\)\(180\)
\(x\)\(=\)\(60\)
x equals 60
Example 4 — Solve with algebra
Corresponding angles measure \((3x+5)^\circ\) and \((5x-25)^\circ\). Find \(x\).
Solution

Corresponding angles are equal, so set the expressions equal.

\(3x+5\)\(=\)\(5x-25\)
\(30\)\(=\)\(2x\)
\(x\)\(=\)\(15\)
x equals 15

Common pitfalls

Co-interior angles are supplementary, not equal. Only corresponding and alternate pairs are equal.
The rules need parallel lines. Without parallelism, the angles have no fixed relationship.
Match the pair type carefully. Interior vs exterior, same side vs alternate — each has its own rule.

Frequently asked questions

What is a transversal?

A line that crosses two or more other lines, creating angles at each intersection.

Which angle pairs are equal for parallel lines?

Corresponding angles and alternate (interior or exterior) angles are congruent.

What are co-interior angles?

Same-side interior angles — between the parallel lines on the same side of the transversal. They are supplementary.

Can these angle relationships prove lines are parallel?

Yes. By the converse, if corresponding or alternate angles are equal (or co-interior angles are supplementary), the lines are parallel.

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