# Angles

## Angle Rules

### Right-angle rule

Straight line angles sum to 180°.

The angles that sub-divide a right angle sum to 90°.

A circle consists of 4 right angles, or 360°. On a graph of a circle, the radial angle enclosed by a right angle is called a quadrant, where quadrant 1 is from 0° to 90°.

Straight lines form pairs of equal and opposite angles
An intersecting straight line through parallel lines will form pairs of equal angles

### Straight-angle rule

A straight angle is the angle of a straight line, which is 180°. The angles that sub-divide a straight line sum to 180°.

When two straight lines intersect, they form two pairs of angles. The pairs are opposite in position and equal in magnitude.

## Pythagoras' Theorem

A famous theorem for calculating the lengths of the sides of a right-angled triangle was invented 2500 years ago by a Greek called Pythagoras. His theorem states:

a2 + b2 = c2, where a and b are the shorter sides of a right-angled triangle with hypotenuse c.

The Pythagoras or Pythagorean Theorem was devised by the Ancient Greek mathematician, Pythagoras (570 - 495 BCE).

For such a famous man, we actually know relatively little about him for sure. However, his legacy as a mathematician and philosopher has had a huge impact on Western culture in many fields. It is a pity most people today just know him for his eponymous triangle theorem, important as it is.

Pythagoras in Raphael's School of Athens

### Trigonometry

Angles of right-angled triangles may be calculated using trigonometric functions. The definitions of the three basic trig functions are:

### Sine

The sine of an angle is equal to the length of the opposite side over the hypotenuse: \$sinB = b/c\$

If two sides of a triangle are known, it is possible to calculate the angle by in the inverse sine function: \$B = sin^{-1}({b/c})\$.

### Cosine

The cosine of an angle is equal to the length of the adjacent side over the hypotenuse: \$cosB = a/c\$

If two sides of a triangle are known, it is possible to calculate the angle by in the inverse cosine function: \$B = cos^{-1}({a/c})\$.

### Tangent

The tangent of an angle is equal to the length of the opposite side over the adjacent side: \$tanB = b/a\$

If two sides of a triangle are known, it is possible to calculate the angle by in the inverse tangent function: \$B = tan^{-1}({b/a})\$.

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