An angle describes the opening between two rays that start at the same point. That common point is the vertex, and the rays are the sides of the angle. Its measure records the amount of turn from one side to the other. Making the drawn sides longer does not change that opening.
What type of angle is it?
An acute angle is greater than 0° and less than 90°. A right angle measures exactly 90°. An obtuse angle lies strictly between 90° and 180°, while a straight angle measures exactly 180°. An angle greater than 180° and less than 360° is reflex. A full turn is 360°, and a zero angle is 0°.
The boundary values matter: 90° is not acute or obtuse, and 180° is not obtuse. Before choosing a name, compare the measure with these boundaries.
How to read a protractor correctly
Place the centre mark over the vertex. Align the zero line with one side of the angle, then follow the other side to the numbered arc. Use the scale that begins at zero on the side you aligned. If a ray is too short, extend its line carefully without changing its direction.
For an opening visibly smaller than a right angle, a reading of 150° instead of 30° usually means you used the wrong scale. A semicircular protractor reads up to 180°. For a reflex angle, measure the remaining smaller angle and subtract it from 360°.
Degrees and radians measure the same turn
Degrees divide a full turn into 360 parts. Radians compare an arc's length with the radius of its circle. The two systems are connected by:
Classifying 120° and converting it to radians
Take an angle of 120°. We will first identify its type and then express the same opening in radians.
Compare 120° with the right and straight angle boundaries. It lies strictly between them, so it is obtuse.
Use the equivalence between 180° and π radians. Multiply 120 by π and divide by 180.
Reduce the fraction by dividing 120 and 180 by 60. Keep π to obtain an exact result.
The angle is obtuse and its radian measure is two thirds of π. The conversion changes the unit, not the opening of the angle.
A right angle is one quarter of a turn and corresponds to half of π radians. When converting a degree measure to radians, multiply by π and divide by 180. Keep the unit clear, especially when checking trigonometric values.
Shared sides do not always mean adjacent angles
Angles can share a vertex and a side while their interiors overlap. Adjacent angles have the shared vertex and side but do not overlap internally. Some courses use the broader word consecutive for angles sharing a vertex and a side. Check the geometric conditions, because terminology varies between curricula.
Two crossing straight lines form pairs of vertically opposite angles. Each opposite pair has equal measures. Neighbouring angles at the crossing add to 180°, which explains why the opposite measures agree.
Relationships based on measure
Congruent angles have equal measures, regardless of orientation or position. Complementary angles total 90°, while supplementary angles total 180°. A 25° angle and a 65° angle are complementary; 120° and 60° are supplementary.
These descriptions can overlap. Two angles may be both adjacent and complementary, or congruent and supplementary. In the latter case each is 90°. Use every marked measure and geometric condition rather than relying on the apparent size of a sketch. Continue with finding complementary angles.
Explore the angle
Move the slider or use the arrow keys. The drawing shows the same angle measure given below.
