Angle Types

What Is an Angle?

An angle is formed when two rays share a common endpoint, called the vertex. We measure angles in degrees, where a full rotation around a point measures 360^{\circ}. Angles are the building blocks of every shape in geometry, so getting comfortable naming and classifying them pays off in every lesson that follows.

Classifying Angles by Size

Every angle falls into one of five categories based on its measure:

• Acute angle — measures between 0^{\circ} and 90^{\circ}. Picture a narrow, pointy wedge, like the hands of a clock at 1:00.
• Right angle — measures exactly 90^{\circ}. This is the corner of a square sheet of paper, often marked with a small square symbol instead of a curved arc.
• Obtuse angle — measures between 90^{\circ} and 180^{\circ}. It looks "wider" than a right angle but hasn't opened all the way into a straight line.
• Straight angle — measures exactly 180^{\circ}. The two rays point in exactly opposite directions, forming a straight line.
• Reflex angle — measures between 180^{\circ} and 360^{\circ}. This is the "outside" sweep you'd get by measuring the larger way around a vertex instead of the smaller way.

Complementary and Supplementary Angles

Two angles are complementary when their measures add up to 90^{\circ}. For example, a 30^{\circ} angle and a 60^{\circ} angle are complementary because 30 + 60 = 90.

Two angles are supplementary when their measures add up to 180^{\circ}. For example, a 110^{\circ} angle and a 70^{\circ} angle are supplementary because 110 + 70 = 180. Supplementary angles often appear as a "linear pair" — two adjacent angles that sit side by side along a straight line, splitting that 180^{\circ} straight angle into two pieces.

Vertical Angles

When two straight lines cross, they form four angles around the intersection point. The pair of angles directly across from each other (not touching, but opposite) are called vertical angles, and a key fact is that vertical angles are always congruent — they have exactly the same measure. Picture an "X" shape: the top and bottom angles of the X are equal to each other, and the left and right angles of the X are equal to each other.

Example

Two lines intersect, forming four angles. One of the angles measures 128^{\circ}. Find the measures of the other three angles.

1. The angle vertical to the 128^{\circ} angle (directly opposite it) must also measure 128^{\circ}, since vertical angles are congruent.
2. The two angles adjacent to the 128^{\circ} angle each form a linear pair with it, so each is supplementary to 128^{\circ}: 180^{\circ} - 128^{\circ} = 52^{\circ}
3. Those two remaining angles are also vertical to each other, confirming they're both 52^{\circ}.

So the four angles around the intersection measure 128^{\circ}, 52^{\circ}, 128^{\circ}, 52^{\circ} going around the point.

Key Takeaways

• Acute: 0^{\circ}\text{-}90^{\circ}, Right: 90^{\circ}, Obtuse: 90^{\circ}\text{-}180^{\circ}, Straight: 180^{\circ}, Reflex: 180^{\circ}\text{-}360^{\circ}.
• Complementary angles sum to 90^{\circ}; supplementary angles sum to 180^{\circ}.
• Vertical angles (formed by two intersecting lines) are always congruent.
• Angles forming a linear pair along a straight line are supplementary.