Section 12.1: Adjacent and Vertical Angles

Property

When two line segments meet at a point, they form an angle. The point where the two sides meet is called the vertex.

• If the sides are less open than a right angle, we call the angle acute.
• If the sides are more open than a right angle, we call the angle obtuse.
• If the two sides are totally open to form a straight line, we call the angle a straight angle.
• In a right angle, the sides are perpendicular.

Examples

• The tip of a slice of pie typically forms an acute angle.
• At 5:00, the hands of a clock form an obtuse angle.
• A perfectly straight pencil lying on a desk represents a straight angle.

Property

We use degrees to measure how open an angle is. The symbol for a degree is $\degree$. A right angle is $90\degree$, and a straight angle is $180\degree$.
The degree measure of an acute angle is less than $90\degree$, and the degree measure of an obtuse angle is between $90\degree$ and $180\degree$. The measure of an angle does not depend on the orientation of the angle, or on how long the sides are. It only depends on how open the sides are.

Examples

• If two angles form a right angle ($90\degree$) and one angle is $35\degree$, the other angle must be $90\degree - 35\degree = 55\degree$.
• An angle that measures $125\degree$ is an obtuse angle because its measure is between $90\degree$ and $180\degree$.
• If two angles form a straight line ($180\degree$) and one is $100\degree$, the other must be $180\degree - 100\degree = 80\degree$.

Explanation

Degrees are like tiny measurement units for angles. A right angle has 90 of them, and a straight line has 180. The more open the angle, the more degrees it measures, regardless of how long its sides are.

Property

Adjacent angles are two angles that share a common vertex and a common side, but have no interior points in common. If angles $\angle ABC$ and $\angle CBD$ are adjacent, they share vertex $B$ and common side $\overrightarrow{BC}$.

Examples