Adjacent Angles
In short: Adjacent angles are neighboring angles that share one vertex and one common side while lying on opposite sides of that side, so they never overlap.
7th GradeGeometry7.G.B.5
What is Adjacent Angles?
Open a laptop halfway and look at the two angles formed on either side of a shared edge: those are adjacent angles. They meet at one point, share one ray, and have no interior space in common.
Three conditions must hold together: a common vertex, a common side, and no overlap. When adjacent angles are put together, their measures simply add, so \(m\angle AOC = m\angle AOB + m\angle BOC\). If the outer sides form a straight line, the pair is called a linear pair and totals \(180^{\circ}\).
How to spot it
- Two angles in a diagram sit next to each other with one ray drawn between them.
- A big angle is split by an inner ray, and you are told the sizes of the parts or the whole.
- The wording says common vertex, common side, or angle addition.
- Rays fan out from one point like slices of a pie.
The method
- Confirm the pair has a shared vertex, a shared ray and no overlapping interior.
- Write the whole angle as the sum of its adjacent parts.
- Substitute known measures, and solve for the missing one by subtracting.
- Check: if the outer rays form a line, the total must be \(180^{\circ}\); if they form a square corner, \(90^{\circ}\).
Adjacent Angles: solved example
Problem
Rays OA, OB and OC leave point O in that order. Angle AOB measures \(48^{\circ}\) and the adjacent angle BOC measures \(37^{\circ}\). Find the measure of angle AOC. If angle AOC were \(130^{\circ}\) and AOB were \(76^{\circ}\), what would BOC be?
Solution
Add the adjacent parts: \(48 + 37 = 85\), so angle AOC is \(85^{\circ}\).
For the second case, subtract: \(130 – 76 = 54\), so angle BOC would be \(54^{\circ}\).
Try it
Question 1
Angles XYZ and ZYW are adjacent, and angle XYW is a straight angle. If XYZ is \(112^{\circ}\), what is ZYW?
Show the answer
A straight angle is \(180^{\circ}\), so \(180 – 112 = 68\) and ZYW is \(68^{\circ}\).
Question 2
Two angles share a vertex and overlap partly. Are they adjacent?
Show the answer
No. Adjacent angles may not overlap; they must lie on opposite sides of their shared ray.
Adjacent Angles vs. Vertical Angles
Adjacent angles are neighbors: they touch along a shared ray, and their measures add together. Vertical angles are opposites at a crossing: they touch only at the vertex and their measures are equal. Ask whether the two angles share a side (neighbors) or face each other across the vertex (opposites).
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Whole dictionary: A–Z · 7th Grade · Geometry