Lesson 54: Angle Relationships

New Concept

When lines intersect, they create predictable angle pairs. Knowing these key definitions allows you to solve for any unknown angle.

Supplementary: Two angles totaling $180^\circ$.
Complementary: Two angles totaling $90^\circ$.
Vertical: Opposite angles (vertical angles) are congruent.

What’s next

This is just the start! Next, you’ll apply these rules in worked examples and see what happens when a line cuts across parallel lines.

Property

Opposite angles formed by two intersecting lines are called vertical angles. Vertical angles are congruent (equal in measure).

Examples

If two lines intersect and one angle is $110^\circ$, the angle vertically opposite to it is also $110^\circ$.
In a cross shape, if the top angle is $75^\circ$, the bottom angle must also be $75^\circ$.
If two intersecting lines form an angle $\angle A = 45^\circ$, its vertical angle partner, $\angle C$, will have the measure $m\angle C = 45^\circ$.

Explanation

Imagine two straight lines crossing like a giant 'X'. The angles directly across from each other are vertical angles. They're perfect mirror images, so they always have the same measure. If you know the size of one angle, you automatically know its opposite twin! It's a fantastic two-for-one deal in the world of geometry.

Property

Supplementary angles are two angles whose measures total $180^\circ$.
Complementary angles are two angles whose measures total $90^\circ$.

Examples

A $130^\circ$ angle and a $50^\circ$ angle are supplementary because $130^\circ + 50^\circ = 180^\circ$.
A $25^\circ$ angle and a $65^\circ$ angle are complementary because $25^\circ + 65^\circ = 90^\circ$.
Adjacent angles on a straight line are always supplementary, like a $120^\circ$ angle and a $60^\circ$ angle.

Explanation

Remember this trick: 'S' in Supplementary stands for 'Straight' line ($180^\circ$), and 'C' in Complementary stands for 'Corner' ($90^\circ$). These angle pairs don't even need to be touching! As long as their measures add up to the magic number, they are considered supplementary or complementary. It's all about teamwork and reaching that total!

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.