Transversals of parallel lines: find angle measures

key notes :

Key Concepts

Parallel Lines and Transversal

• Parallel Lines: Two lines that never meet, no matter how far they are extended.
• Symbol: l∥m
• Transversal: A line that intersects two or more lines at distinct points.

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Types of Angles Formed by a Transversal

When a transversal cuts two parallel lines, the following types of angles are formed:

Corresponding Angles (∠)

• Angles in the same relative position at each intersection.
• Property: Corresponding angles are equal.
• Example: ∠1=∠5

Alternate Interior Angles

• Angles on opposite sides of the transversal, inside the parallel lines.
• Property: Alternate interior angles are equal.
• Example: ∠3=∠6

Alternate Exterior Angles

• Angles on opposite sides of the transversal, outside the parallel lines.
• Property: Alternate exterior angles are equal.
• Example: ∠1=∠8

Interior Angles on the Same Side (Consecutive Interior Angles / Co-Interior Angles)

• Angles on the same side of the transversal, inside the parallel lines.
• Property: Sum = 180^∘ (Supplementary)
• Example: ∠3+∠5=180^∘

Exterior Angles on the Same Side

• Angles on the same side of the transversal, outside the parallel lines.
• Property: Sum = 180^∘

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Steps to Find Angle Measures

1. Identify the Parallel Lines and Transversal.

2. Label the angles formed at intersections.

3. Use angle relationships:

• Corresponding angles → equal
• Alternate interior angles → equal
• Alternate exterior angles → equal
• Co-interior angles → sum = 180°

4 .Write equations if needed.

5. Solve for unknown angles using algebra.

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Examples

Example 1:
Two parallel lines l∥m are cut by a transversal. One angle is 70^∘ Find all other angles.

Solution:

• Corresponding angle =70^∘
• Alternate interior angle = 70^∘
• Co-interior angle = 180−70=110^∘
• Alternate exterior angle = 70^∘
• Exterior angle on same side = 110^∘

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Example 2:

Two parallel lines are cut by a transversal. One angle is 2x+10^∘ and its corresponding angle is 70^∘ Find x.

Solution:

Corresponding angles are equal:

2x+10=70

2x=60

x=30

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Tips and Tricks

• Always check if lines are parallel; the angle relationships only apply for parallel lines.
• Start by identifying known angles and using equality or supplementary relationships.
• Use algebra carefully for unknown angles.
• Draw a clear diagram; labeling makes calculations easier.

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Practice Problems

1. Two parallel lines are cut by a transversal. One angle measures 55^∘ Find all other angles.
2. One angle formed by a transversal is 3x−10^∘, its alternate interior angle is 50^∘ Find x.
3. Find the unknown angles if two parallel lines are intersected by a transversal and one angle is 120^∘

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Learn with an example

Let’s Practice!