Angles

Supplementary Angles

 http://www.mathsisfun.com/geometry/supplementary-angles.html

Two Angles are Supplementary if they add up to 180 degrees.

These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°. Notice that together they make a straight angle.
But the angles don't have to be together. These two are supplementary because 60° + 120° = 180°

	If the two angles add to 180°, we say they "Supplement" each other. Supplement comes from Latin supplere, to complete or "supply" what is needed.
	Spelling: be careful, it is not "Supplimentary Angle" (with an "i")

Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°
How can you remember which is which? Easy! Think:

• "C" of Complementary stands for "Corner" (a Right Angle), and
• "S" of Supplementary stands for "Straight" (180 degrees is a straight line)
  

You could also think "Supplement" (like a Vitamin Supplement) is something extra, so it it bigger.

 

 

Exterior Angles of Polygons

http://www.mathsisfun.com/geometry/exterior-angles-polygons.html

The Exterior Angle is the angle between any side of a shape,
and a line extended from the next side.

Note: when you add up the Interior Angle and Exterior Angle you get a straight line, 180°. (See Supplementary Angles)

Polygons

A Polygon is any flat shape with straight sides

The Exterior Angles of a Polygon add up to 360°
		In other words the exterior angles add up to one full revolution (Exercise: try this with a square, then with some interesting polygon you invent yourself.) Note: This rule only works for simple polygons

Here is another way to think about it: Each lines changes direction until you eventually get back to the start:

 

Interior Angles of Polygons

http://www.mathsisfun.com/geometry/interior-angles-polygons.html

An Interior Angle is an angle inside a shape.

Triangles

The Interior Angles of a Triangle add up to 180°

90° + 60° + 30° = 180°	80° + 70° + 30° = 180°
It works for this triangle!	Let's tilt a line by 10° ... It still works, because one angle went up by 10°, but the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

90° + 90° + 90° + 90° = 360°	80° + 100° + 90° + 90° = 360°
A Square adds up to 360°	Let's tilt a line by 10° ... still adds up to 360°!
The Interior Angles of a Quadrilateral add up to 360°

Because there are Two Triangles in a Square

The interior angles in this triangle add up to 180° (90°+45°+45°=180°)

... and for this square they add up to 360° ... because the square can be made from two triangles!

Pentagon

		A pentagon has 5 sides, and can be made from three triangles, so you know what ... ... its interior angles add up to 3 × 180° = 540° And if it is a regular pentagon (all angles the same), then each angle is 540° / 5 = 108° (Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)
	The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

			If it is a Regular Polygon (all sides are equal, all angles are equal)
Shape	Sides	Sum of Interior Angles	Shape	Each Angle
Triangle	3	180°		60°
Quadrilateral	4	360°		90°
Pentagon	5	540°		108°
Hexagon	6	720°		120°
Heptagon (or Septagon)	7	900°		128.57...°
Octagon	8	1080°		135°
Nonagon	9	1260°		140°
...	...	..	...	...
Any Polygon	n	(n-2) × 180°		(n-2) × 180° / n

So the general rule is:

Sum of Interior Angles = (n-2) × 180°

Each Angle (of a Regular Polygon) = (n-2) × 180° / n

Perhaps an example will help:

Example: What about a Regular Decagon (10 sides) ?

Sum of Interior Angles = (n-2) × 180°   = (10-2)×180° = 8×180° = 1440° And it is a Regular Decagon so: Each interior angle = 1440°/10 = 144°

 

 

Exterior Angle

The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.

Interior Angle

An Interior Angle is an angle inside a shape.

http://www.mathsisfun.com/geometry/interior-angles.html

Transversals

http://www.mathsisfun.com/geometry/transversal.html

A Transversal is a line that crosses at least two other lines.

                                         The red line is the transversal in each example:

Transversal crossing two lines	this Transversal crosses two parallel lines	... and this one cuts across three lines