📐 Review: What Is an Angle?
An angle is formed when two rays share a common endpoint called the vertex. We measure angles in degrees (°).
| Angle Type | Measure |
|---|---|
| Acute | Less than 90° |
| Right | Exactly 90° |
| Obtuse | Between 90° and 180° |
| Straight | Exactly 180° |
🎯 What's Next?
Now we'll study special pairs of angles — angles that have a specific relationship to each other. We'll focus on two important types:
- Adjacent Angles (angles next to each other)
- Vertical Angles (angles across from each other)
📏 Part 1: Adjacent Angles
📖 Definition: Adjacent Angles
Adjacent angles are two angles that:
- Share a common vertex
- Share a common side (ray)
- Their other sides form a straight line (opposite rays)
In other words, adjacent angles are "next to each other" and together they form a straight angle (180°).
In the diagram to the right, a ray OB divides a straight angle into two parts:
- $\angle AOB$ and $\angle BOC$ are adjacent angles
- They share vertex O and common side OB
- Their other sides OA and OC form a straight line
Interactive visualization — available in the full lesson
⭐ Property of Adjacent Angles
The sum of adjacent angles equals 180°
$$\angle AOB + \angle BOC = 180°$$
(when $OA$ and $OC$ form a straight line)
This is the fundamental property of adjacent angles. If you know one angle, you can always find the other!
📝 Proof: Why Do Adjacent Angles Sum to 180°?
| Step | Statement | Reason |
|---|---|---|
| 1 | $\angle AOB$ and $\angle BOC$ are adjacent angles | Given |
| 2 | $OA$ and $OC$ are opposite rays | Definition of adjacent angles |
| 3 | $\angle AOC$ is a straight angle | Opposite rays form a straight angle |
| 4 | $\angle AOC = 180°$ | A straight angle measures 180° |
| 5 | $\angle AOB + \angle BOC = \angle AOC$ | Angle Addition Postulate |
| 6 | $\angle AOB + \angle BOC = 180°$ | Substitution (steps 4 & 5) ✓ |