Kinds of Angles
An acute angle is between 0 and 90 degrees. A right angle is 90 degrees, an obtuse angle is between 90 and 180, a straight angle is 180 and a full angle is 360. Complementary angles add up to 90 degrees and supplementary angles add up to 180. Vertically opposite angles are equal. Angles at a point add up to 360 degrees.
Parallel Lines
When a transversal cuts two parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles on the same side add up to 180 degrees.
Triangles
The three angles of a triangle add up to 180 degrees. An exterior angle equals the sum of the two opposite interior angles. Any two sides add up to more than the third side. In an equilateral triangle every angle is 60 degrees. In an isosceles triangle, angles opposite equal sides are equal. The three medians meet at the centroid, which divides each median in the ratio 2 to 1.
Pythagoras
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Common triples are 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25; and 20, 21, 29.
Areas
The area of a triangle is half the base times the height. Heron's formula uses s equal to half the perimeter, and the area is the square root of s times s minus a times s minus b times s minus c. An equilateral triangle of side a has area root 3 over 4 times a squared and height root 3 over 2 times a. For similar triangles, the ratio of areas is the square of the ratio of sides.
Polygons
The interior angles of an n-sided polygon add up to n minus 2 times 180 degrees. The exterior angles always add up to 360. Each exterior angle of a regular polygon is 360 over n. The number of diagonals is n times n minus 3 over 2.
Shortcuts
If the exterior angle of a regular polygon is given, n equals 360 divided by it. For angle ratios, divide 180 by the sum of the ratio parts. Memorise Pythagorean triples.
Worked Examples
Example 1: Two angles of a triangle are 50 and 60 degrees. Find the third.
Solution: 180 minus 110 equals 70 degrees.
Example 2: Find each interior angle of a regular hexagon.
Solution: Six minus 2 times 180 divided by 6 equals 120 degrees.
Example 3: Find the area of a triangle with sides 13, 14 and 15.
Solution: s is 21. Area is the square root of 21 times 8 times 7 times 6, which is 84.
Quick Check
Question 1: What do the exterior angles of any polygon sum to?
Answer: 360 degrees.
Question 2: How many diagonals does an octagon have?
Answer: Twenty.
Question 3: What ratio does the centroid divide a median in?
Answer: 2 to 1.