Lesson 11-3: Understand Angle Relationships and Parallel Lines

Property

A transversal is a line that intersects two or more other lines at distinct points. When a transversal intersects two lines, it creates eight angles at the two intersection points.

Examples

Property

When two lines intersect at a point, four angles are formed.
Angles on a straight line are supplementary, and the sum of their measures is $180^\circ$.
Angles opposing each other at a vertex are called vertical angles, and they are equal in measure.

Examples

• Two lines intersect. One angle is $40^\circ$. The angle opposite (vertical) to it is also $40^\circ$. The angles adjacent (supplementary) to it are each $180^\circ - 40^\circ = 140^\circ$.
• In an intersection, an angle $\angle A$ and an angle $\angle B$ are supplementary. If the measure of $\angle A$ is $110^\circ$, then the measure of $\angle B$ is $180^\circ - 110^\circ = 70^\circ$.
• Two intersecting lines form four angles. If one angle is a right angle ($90^\circ$), its vertical angle is also $90^\circ$, and its supplementary angles are also $180^\circ - 90^\circ = 90^\circ$. All four angles are right angles.

Explanation

Think of an 'X'. Angles side-by-side on a straight line are 'supplements' that complete a half-circle ($180^\circ$). Angles across from each other are 'vertical' and are perfect mirror images, so they must be equal.

Property

When a transversal cuts two parallel lines, corresponding angles are on the same side of the transversal and on the same side of each parallel line. Corresponding angles are congruent.

Examples

• If a transversal cuts two parallel lines and the top-left angle is $115^\circ$, the bottom-left angle is also $115^\circ$.
• Given parallel lines cut by a transversal, if $\angle 2$ is $65^\circ$, its corresponding angle $\angle 6$ is also $65^\circ$.
• If the top-right angle is $\angle 1$ and the bottom-right angle is $\angle 5$, then $m\angle 1 = m\angle 5$.

Explanation

Imagine sliding the top group of four angles down the transversal until it sits on top of the bottom group. The angles that perfectly match up are corresponding angles! They hold the same position at each intersection—like they are both in the 'top-right' spot. Since parallel lines have the same direction, these corresponding angles are always identical.