Key takeaway: An angle is the figure formed by two rays that start at the same vertex, and its measure tells how far one ray must turn to land on the other.
Grade 10 – GeometryG-CO.A.1Geometry and Measurement
Definition of Angle
An angle forms when two rays leave one common point, the vertex. The rays are the sides. How wide they open is the measure, usually given in degrees.
One full turn is \(360^{\circ}\). Openings narrower than a square corner, under \(90^{\circ}\), are called acute. A square corner is a right opening at exactly \(90^{\circ}\). Wider ones up to \(180^{\circ}\) are obtuse, and exactly \(180^{\circ}\) is a straight opening. A protractor reads the measure when its center sits on the vertex.
Formula and notation
Use three letters with the vertex in the middle, \(\angle ABC\), or just the vertex letter when nothing is ambiguous. The measure carries an \(m\):
\[ m\angle ABC = 64^{\circ} \]
Equal measures make the openings congruent, written with the symbol \(\cong\).
Key facts
- The vertex letter always goes in the middle of a three-letter name.
- Size depends on the opening between the sides, not on how long they are drawn.
- Openings that fill a straight line add up to 180 degrees.
- Two neighbors sharing a vertex and one side, with no overlap, are called adjacent.
Angle example
Two neighbors fill a straight line. One measures \((4x + 7)^{\circ}\) and the other measures \((2x + 17)^{\circ}\). What is the size of each?
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A straight line is \(180^{\circ}\), so \((4x + 7) + (2x + 17) = 180\).
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Combine: \(6x + 24 = 180\), so \(6x = 156\) and \(x = 26\).
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First: \(4(26) + 7 = 111\). Second: \(2(26) + 17 = 69\).
The measures are \(111^{\circ}\) and \(69^{\circ}\), and \(111 + 69 = 180\).
Common mistake
Watch out
Ravi compares two drawings and says the one with longer sides is bigger, since its sides reach farther across the page.
Why it is wrong
The sides go on forever; the drawn length is only a sketch. What counts is the opening. A \(40^{\circ}\) opening drawn with 10-inch sides is still smaller than a \(70^{\circ}\) opening drawn with 1-inch sides.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement