Special Parallelograms

A parallelogram may be equiangular (rectangle), equilateral (rhombus) or both equiangular and equilateral. An example of a special parallelogram that is both equiangular and equilateral is our friend the square.

Rectangle

A rectangle is a parallelogram with 4 right angles. A rectangle has the following rules:
(1) All the rules of a parallelogram.
(2) Four right angles. Remember that a right angle measures 90 degrees.
(3) Diagonals which are congruent (they have the same length).
The picture of rectangle ABCD below shows all three rules listed above.

Sample:
In rectangle ABCD below, diagonals AC and BD intersect at point R. If AR = 2x - 6 and CR = x + 10, find BD.

Since the diagonals of a rectangle bisect each other, we can say that
AR = CR.
We equate the values of AR and CR and solve for x.
2x - 6 = x + 10
2x - x = 10 + 6
x = 16
Use either of the given equations to determine that each segment equals 26. Since they are all equal, BD = 26.

Rhombus

A rhombus is a parallelogram with 4 congruent or equal sides. A rhombus has the following rules:
(1) All the rules of a parallelogram.
(2) Four sides that have the same length.
(3) Diagonals that intersect at right angles.
(4) Diagonals that bisect opposite pairs of angles.

Sample:
Given that ABCD is a rhombus and the measure of angle D = 60 degrees.
Find the measure of angles A and B.

Solution:
Triangle ABC is isosceles since line segment AB is congruent to line segment BC. Then we can say that the base angles of triangle ABC must be congruent or equal. Since we know the diagonal bisects the angles A and C, we must have two congruent triangles here. If so, then the measure of angle D = the measure of angle B = 60 degrees.

In triangle AEB, angle AEB is a right triangle because the diagonals of a rhombus are perpendicular to each other. Since the sum of the degree measures of the angles of a triangle is 180 degrees, we can say that the measure of angle A must be 30 degrees. How do we get 30?

Measure of angle A = 180 - 90 degrees - 60 degrees
Measure of angle A = 30 degrees

Square

A square is a parallelogram with 4 right angles and 4 sides that have the same length. A square has all the rules of a rectangle and a rhombus as shown in square ABCD below.

Sample:
In square ABCD, AB = x + 4. What is the perimeter of square ABCD?
Solution:
A square has the same length on all 4 sides.
We can use the formula P = side times 4, or P = 4s, where P = perimeter and s = side of square.
P = 4s
P = 4(x + 4)....We apply the distributive rule here and get
P = 4x + 16
Our perimeter is 4x + 16.

Exterior Angles of Polygons

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

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°

Note: Interior Angles are sometimes called "Internal Angles"