Angles

Angle Basics

Angles are a measure of turn and are measured in degrees.
A full turn is 360 degrees.
A half turn is 180 degrees.
A quarter turn is 90 degrees, also known as a right angle.
Acute angles are less than 90 degrees.
Obtuse angles are more than 90 degrees but less than 180 degrees.
Straight angles are exactly 180 degrees and form a straight line.

Types of Angles

Adjacent angles are two angles that have a common side and a common vertex (the shared endpoint of the sides).
Vertically opposite angles occur when two lines intersect. They are located across from each other at the intersection point and are always equal.
Complementary angles are two angles that add up to 90 degrees.
Supplementary angles are two angles that add up to 180 degrees, i.e., they form a straight line.
Corresponding angles are pairs of angles that are in the same relative position at an intersection of a transversal and at least two lines. If the two lines are parallel, the corresponding angles are equal.
Alternate angles are pairs of angles formed when a straight line crosses two others. If these two lines are parallel, the alternate angles are equal.

Angle Properties

The angles in a triangle always add up to 180 degrees.
The angles in a quadrilateral always add up to 360 degrees.
The exterior angle of a triangle is equal to the sum of the two opposing interior angles.
On a straight line, the sum of the angles is always 180 degrees.
Around a point, the sum of the angles is 360 degrees.
The interior angles of a polygon can be calculated using the formula (n-2) * 180, where n is the number of sides.

Circle Geometry

A circle has 360 degrees.
Tangent to a circle forms a right angle with the radius at the point of contact.
Angles in a semicircle are always 90 degrees.
Angles at the centre of a circle are twice the angle at the circumference when subtended by the same arc.
Alternate segment theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.
Cyclic quadrilaterals: Opposite angles in a cyclic quadrilateral add up to 180 degrees.
The angle in a semicircle is always a right angle.
The angle subtended at the centre of the circle is twice the angle subtended at the circumference by the same arc.
Angles subtended by the same arc or chord are equal.

Each of these sections and points highlight essential points about angles that are crucial for understanding and solving geometry problems. This knowledge will be particularly useful when dealing with questions surrounding angles in polygons, triangles, and circular geometries.