Key takeaway: Adjacent angles are neighbors in a fan of rays: they meet at one vertex, lean on one shared ray, and never overlap, so their measures add.
Grade 77.G.B.5Geometry and Measurement
Definition of Adjacent Angles
Adjacent angles are neighbors. They meet at the same corner point, called the vertex, and one ray lies between them as a shared wall. Neither angle overlaps the other, so you can add their measures to get the size of the combined angle.
Three conditions must hold together: one vertex, one shared ray, and no overlap. Think of two slices of a pizza that sit side by side. Their tips touch at the center, the cut between them is shared, and each slice has its own crust. Adjacent angles can be any size, and their total can be less than, equal to or greater than \(90^{\circ}\).
Formula and notation
If \(\vec{BC}\) is the shared side of \(\angle ABC\) and \(\angle CBD\), the whole angle is their sum:
\[m\angle ABD = m\angle ABC + m\angle CBD\]
This is the angle addition idea: parts add to the whole.
Key facts
- Drop any one of the three conditions and the pair stops counting as neighbors.
- Neighbors may total 90 degrees, 180 degrees or any other amount.
- A pair of neighbors lying along a straight line is called a linear pair.
- Opposite angles at a crossing are never neighbors, since they meet only at the vertex.
Adjacent Angles example
Angle \(ABC\) measures \((x + 20)^{\circ}\) and the angle \(CBD\) next to it measures \((2x – 5)^{\circ}\). Together they make angle \(ABD\) of \(105^{\circ}\). Find \(x\) and each angle.
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The parts add to the whole: \((x + 20) + (2x – 5) = 105\).
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Combine: \(3x + 15 = 105\), so \(3x = 90\) and \(x = 30\).
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Angle \(ABC = 30 + 20 = 50\) degrees; angle \(CBD = 2(30) – 5 = 55\) degrees.
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Check: \(50 + 55 = 105\).
\(x = 30\); the angles measure \(50^{\circ}\) and \(55^{\circ}\).
Common mistake
Watch out
Two angles share the ray \(\vec{PQ}\), but one has its vertex at \(P\) and the other has its vertex at \(Q\). Luis calls them adjacent because they “touch along a ray.”
Why it is wrong
The vertex must be the same point for both angles. Angles with different vertices cannot be neighbors even if they lie along the same ray. Check the three conditions every time: one vertex, one shared side, and no overlap.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement