ANGLES, LINES
Lines AB and CD are parallel to one another (hence the » on the lines).
a and d are known as vertically opposite angles. Vertically opposite
angles are equal. (b and c, e and h, f and g are also vertically opposite).
g and c are corresponding angles. Corresponding angles are equal. (h
and d, f and b, e and a are also corresponding).
d and e are alternate angles. Alternate angles are equal. (c and f are
also alternate). Alternate angles form a 'Z' shape and are sometimes
called 'Z angles'.
a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d
and c, c and a, d and b, f and e, e and g, h and g, h and f are also
adjacent).
d and f are interior angles. These add up to 180 degrees (e and c are
also interior).
Any two angles that add up to 180 degrees are known
as supplementary angles.
Angle Sum of a Triangle
We know that x, y and z together add up to 180 degrees, because these
together is just the angle around the straight line. So the three angles
in the triangle must add up to 180 degrees.
Angle Sum of a Quadrilateral
A quadrilateral is a shape with 4 sides.
Now that we know the sum of the angles in a triangle, we can work out
the sum of the angles in a quadrilateral.
For any quadrilateral, we can draw a diagonal line to divide it into two
triangles. Each triangle has an angle sum of 180 degrees. Therefore the
total angle sum of the quadrilateral is 360 degrees.
Exterior Angles
The exterior angles of a shape are the angles you get if you extend the
sides. The exterior angles of a hexagon are shown:
polygon is a shape with straight sides. All of the exterior angles of a
polygon add up to 360°.
Therefore if you have a regular polygon (in other words, where all the
sides are the same length and all the angles are the same), each of the
exterior angles will have size 360 ÷ the number of sides. So, for
example, each of the exterior angles of a hexagon are 360/6 = 60°.
Exterior Angle of a Triangle
Angle x is an exterior angle of the triangle:
The exterior angle of a triangle is equal to the sum of the interior angles
at the other two vertices. In other words, x = a + b in the diagram.
A parallelogram has two pairs of equal sides.
It has two pairs of equal angles.
The opposite sides are parallel.
The diagonals bisect each other.
A rhombus has four sides of equal lengths.
It has two pairs of equal angles.
The opposite sides are parallel.
The diagonals bisect each other at right angles.
A kite has two pairs of equal sides.
It has one pair of equal angles.
The diagonals bisect at right angles.
The sum of interior angles in a quadrilateral is 360°.
Exercise 1.
The formula for calculating the sum of interior angles is:
(n-2) x 180o (where n is the number of sides)
(5-2) x 180 = 900-360= 540
Exercise 2.
Calculate the sum of interior angles in an octagon.
(8-2) x 180 = 1080
Exercise 3.
Calculating the exterior angles of regular polygons
The formula for calculating the size of an exterior angle is:
Exterior angle of a polygon = 360 divided by number of
sides.
Remember the interior and exterior angle add up to 180°.
Exercise 4.
Calculate the size of the exterior and interior angle in a
regular pentagon.
Exercise 5.
Exercise 6.
(n-2)x180 = 540
33+140+2x+x+x+75=540
248+4x=540
4x=540-248
4x=292
X= 73
Exercise 7.
