Angle
In short: An angle is the figure formed by two rays that share a common endpoint, and its measure in degrees tells how far one ray must turn to reach the other.
GeometryGeometryG-CO.A.1
What is Angle?
An angle forms where two rays leave the same point, called the vertex. The rays are its sides. The measure says how much you rotate one side to land on the other, counted in degrees, with a full turn equal to \(360^{\circ}\).
Sizes are sorted by measure: acute below \(90^{\circ}\), right at exactly \(90^{\circ}\), obtuse between \(90^{\circ}\) and \(180^{\circ}\), and straight at \(180^{\circ}\). A name uses three letters with the vertex in the middle, as in \(\angle ABC\).
How to spot it
- A diagram has two segments meeting at a corner marked with a small arc or a tiny square.
- The question gives a protractor reading, a turn such as a quarter turn, or a corner of a polygon.
- You see the symbol \(\angle\) followed by three letters, or a single letter written at the vertex.
- A problem asks whether a corner is sharp, square or wide, or asks for its size in degrees.
The method
- Find the vertex and the two sides, then name the corner with the vertex letter in the middle.
- Place the protractor center on the vertex and line one side up with the zero mark.
- Read the number where the other side crosses the scale that began at zero.
- Classify the result: below 90 is acute, exactly 90 is right, between 90 and 180 is obtuse.
Angle: solved example
Problem
The edge of a laptop screen crosses the 164° mark on a protractor, and the edge of the keyboard deck crosses the 31° mark, both measured from the same zero line. Find the opening \(\angle ABC\) and classify it.
Solution
Subtract the two readings: \(164 – 31 = 133\), so \(m\angle ABC = 133^{\circ}\).
Since \(90 < 133 < 180\), the opening is obtuse, 133°.
Try it
Question 1
A corner measures \(67^{\circ}\). Classify it and find how many more degrees it needs to become a right corner.
Show the answer
It is below \(90^{\circ}\), so it is acute. It needs \(90 – 67 = 23\) more degrees.
Question 2
At 4:00 on a clock face, what is the smaller turn from the minute hand to the hour hand?
Show the answer
Each hour mark is \(360 \div 12 = 30^{\circ}\) apart, and 4 marks separate them: \(4 \times 30 = 120^{\circ}\).
Angle vs. Adjacent Angles
A single angle is one opening between two rays. Adjacent angles are a pair of openings that share a vertex and one side without overlapping, like two slices of a pie. If their outer sides form a straight line, the two measures add to \(180^{\circ}\).
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