1. GEOMETRY

2. ANGLE STATEMENTS
Remember: You must supply a geometrical reason when calculating angles!
Adjacent Angles On A Straight Line Add To 180°
1. Find x
x + 37 = 180
(adj. ’s on a str. line = 180°)
- 37
- 37 x
37°
x = 143°
2. Find x
x = 180
(adj. ’s on a str. line = 180°)
- 119
- 119
119° x
x = 61°

3. Complementary Angles Add To 90°
When two angles make up a right angle (i.e. 48° and 42° are complementary angles)
e.g. Find x
x + 50 = 90
(complementary angles)
- 50
- 50 x
50°
x = 40°
(therefore 40° is the complement of 50°)
Supplementary Angles Add To 180°
When two angles make up a straight angle (i.e. 125° and 55° are supplementary angles)
e.g. What angle is the supplement of 10°?
x + 10 = 180
(supplementary angles)
- 10
- 10
x = 170°
(therefore 170° is the supplement of 10°)

4. Vertically Opposite Angles Are Equal
Vertically opposite angles are formed by two straight lines
1. Find x
2. Find x
12°
38° x
58° x
x = 58°
(vert. opp. ’s are =)
x =
(vert. opp. ’s are =)
x = 50°
Angles At A Point Add To 360°
1. Find x
2. Find x
x = 360
82°
(’s at a point = 360°) x
34°
x = 360
71° x
59°
- 302
- 302
x + 34 = 360
(’s at a point = 360°)
x = 58°
- 34
- 34
x = 326°

5. Interior Angles In A Triangle Add To 180°
1. Find x
2. Find x x
85°
46° x
52°
x = 180
x = 180
(’s in a triangle add to 180°)
(’s in a triangle add to 180°)
x = 180
x = 180
- 137
- 137
- 136
- 136
x = 43°
x = 44°
Base Angles In An Isosceles Triangle Are Equal
1. Find x
x + x = 180
(base ’s of an isosceles triangle)
2x = 180
(’s in a triangle add to 180°)
- 40
- 40
40°
2x = 140
÷ 2
÷ 2 x
x = 70°

6. Exterior Angles Of A Polygon Add To 360°
1. Find x
2. Find x in this regular pentagon
68°
Regular means equal sides and angles
55° x
56°
55°
68° x
x = 360
5x = 360
(ext. ’s of a polygon add to 360°)
(ext. ’s of a polygon add to 360°)
x = 360
÷ 5
÷ 5
- 298
- 298
x = 72°
x = 62°
3. A regular polygon has exterior angles of 36°. How many sides does it have?
36x = 360
The number of sides = the number of angles
÷ 36
÷ 36
x = 10

7. The Sum Of The Interior Angles Of A Polygon Is (n – 2)  180
n = number of sides of a polygon
1. Find the angle sum of this regular hexagon
2. Find x
72° x
divide into triangles from one corner
n = 6 (or 4 triangles)
n = 5 or 3 triangles
Interior angle sum = (6 – 2) x 180
Interior angle sum = (5 – 2) x 180
(interior angle sum of a polygon)
(interior angle sum of a polygon)
Interior angle sum = 720°
Interior angle sum = 540°
x = 540 ÷ 5
x = 108°
Another method is to calculate an exterior angle first then use adjacent angles on a straight line to calculate interior angle
Exterior angle = 72°
x + 72 = 180
(adjacent. angles on a straight line = 180°)
x = 108°

8. Perpendicular Lines Parallel Lines - Always cross at right angles.
e.g. B
AB is perpendicular to CD or AB  CD C D A
Parallel Lines
- Never meet and are always the same distance apart.
e.g. E A B
AB is parallel to CD or AB ⁄⁄ CD C D
EF is known as a transversal F

9. Angle Statements and Parallel Lines
Alternate Angles On Parallel Lines Are Equal
- There are two pairs of alternate angles between parallel lines and a transversal.
e.g.
e.g. Find x
113° x
x = 113°
(Alternate angles on parallel lines are equal)
Corresponding Angles On Parallel Lines Are Equal
- There are four pairs of corresponding angles between parallel lines and a transversal.
e.g. Find x
e.g.
x = 122°
122°
(Corresponding angles on parallel lines are equal) x

10. Co-Interior Angles On Parallel Lines Add To 180
- There are two pairs of co-interior angles between parallel lines and a transversal.
e.g.
e.g. Find x x
77°
x + 77 = 180
(Co-interior angles in parallel lines add to 180°)
- 77
- 77
x = 103°
Remember to always add ‘on parallel lines’ with your angle statements

11. Similar Triangles And Other Shapes
- One shape is similar to another if they have exactly the same shape. The ratios of the corresponding sides are therefore the same.
- Triangles are similar if they have the same angles
e.g. The following two triangles are similar. Work out the lengths x and y B F
First calculate ratio between corresponding sides x 4 y 20
AC = 15
EG 6 E G A 6 C
= 2.5 15
To find x we need to multiply the corresponding side by the ratio:
To find y we need to divide the corresponding side by the ratio:
x = 4 × 2.5
y = 20 ÷ 2.5
= 10
= 8

12. Parts Of A Circle Radius Diameter Chord Arc Sector Segment
Circumference
Tangent

13. Angle Properties of Circles
Base Angles Of An Isosceles Triangle Are Equal
Because two sides of the triangle are radii, an isosceles triangle is formed
e.g.
40° x r r
x = 40°
(base ’s of an isosceles triangle)
The Angle At Centre Is Twice The Angle At The Circumference
e.g. Find x
Proof:
x = 180 – 2A
x = 2 × 42
x + C = 180
42° A B
( at centre = 2 ×  at circumf.)
180 – 2A + C = 180 x
C = 2A A x C D
D = 2B
x = 84°
C + D = 2A + 2B
C + D = 2(A + B)

14. Angle In A Semi Circle Is A Right Angle
- This case is a special version of the previous rule
e.g.
x = 90°
( in a semi-circle) x
Angles On The Same Arc Are Equal
e.g. Find x:
Proof:
C = 2A
x = 32°
C = 2B
(’s on the same arc)
2A = 2B C
A = B
32° A x B
There are 2 arcs joining angles

15. The Angle Between Tangent And Radius Is A Right Angle
e.g.
x = 90°
(tangent  radius) x
If Two Tangents Are Drawn From A Point To A Circle They Are The Same Length
e.g.
2x = 180
( sum isos. triangle)
54°
- 54
- 54
2x = 126
÷ 2
÷ 2 y
x = 63° x
y + 63 = 90
(tangent  radius)
- 63
- 63
y = 27°

16. Cyclic Quadrilaterals
- Are four sided figures with all four vertices (corners) lying on the same circle.
Opposite Angles Of A Cyclic Quadrilateral Add To 180
e.g.
Proof:
x + 79 = 180 x
(opp. ’s, cyc. quad)
2A + 2B = 360
- 79
- 79 2B
A + B = 180 B 2A A
x = 101°
79°
Exterior Angle Of A Cyclic Quadrilateral Equals Opposite Interior Angle
e.g.
Proof:
A + B = 180 A
110°
x = 110°
B + C = 180
B = 180 – C
(ext. , cyc. quad)
A – C = 180 B
A = C x C