GEOMETRY

Lines and Angles

Right Angles

- Are 90° or a quarter turn through a circle e.g.

1) Acute:

Angle Types

- Angles can be named according to their sizes

2) Obtuse:

3) Reflex:

- Straight lines are 180°

Are angles less than 90° Are angles between 90° and 180°

Are angles between 180° and 360°

Naming Angles

- Where the rays meet (vertex) gives the middle letter of the angle name

e.g. Name the following angles

a) b)

A

B

C

ZY

X

ABC or CBA

XYZ or ZYX

Measuring Angles

e.g. Measure the following angles

a) b)

A

B

C

ABC =

43°

Make sure you read from the scale starting from 0 on the line!

ZY

X

XYZ =

141°

- When measuring reflex angles, measure smaller angle and subtract from 360°

e.g. Measure the following angle

A

B

C

ABC =

360 - 65 = 295°

Drawing Angles

e.g. Draw the following angles

a) b)ABC = 62°

XYZ = 156°

A B

C

XY

Z

- When drawing reflex angles, subtract angle from 360° and draw this new angle

e.g. Draw the following angle

a) ABC = 274°

360 - 274= 86

A B

C

Don’t forget to add an arc to show the correct angle!

Add the arc to the outside to indicate a reflex angle

Estimating Angles

- Involves guessing how big an angle is- Firstly decide whether the angle is acute, obtuse or reflex

e.g. Estimate the size of the following angles

A

B

C

45° ABC =

a) b)

ZY

X

XYZ =

220°

Special Triangles

Equilateral Triangle- All sides are equal- All angles are equal(60°)

Isosceles Triangle- Two sides are equal- Base angles are equal

Right Angle Triangle- Contains a 90° angle

Scalene Triangle- No sides are equal- No angles are equal

Describing Triangles By Angles

1) Acute Triangle:

2) Right Angle Triangle:

3) Obtuse Triangle:

All angles are less than 90°

Contains a 90° angle

Contains an angle greater than 90°

Quadrilaterals

- Are four sided figures- Have two diagonals

Diagonals go between opposite corners

Special Spaces and their Properties

Square Rectangle/Oblong Parallelogram

Kite/Diamond Arrowhead Trapezium

Isosceles Trapezium Rhombus

Dashes on lines indicate equal length and arrows indicate parallel lines

ANGLE STATEMENTSANGLE STATEMENTS

Remember: You must supply a geometrical reason when calculating angles!

Adjacent Angles On A Straight Line Add To 180°

x 37°

x + 37 = 180 (adj. ’s on a str. line = 180°)- 37 - 37

x = 143°

x119°

x + 119 = 180 (adj. ’s on a str. line = 180°)- 119 - 119

x = 61°

1. Find x

2. Find x

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

x50°

x + 50 = 90 (complementary angles)- 50 - 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°)

Vertically Opposite Angles Are EqualVertically opposite angles are formed by two straight lines

1. Find x

x 58°

x = 58° (vert. opp. ’s are =)

2. Find x

x

38° 12°

x = 38 + 12 (vert. opp. ’s are =)x = 50°

Angles At A Point Add To 360°

1. Find x

34°x

x + 34 = 360 (’s at a point = 360°)- 34 - 34

x = 326°

2. Find x

x71°59°

82°x + 90 + 82 + 71 + 59 = 360 (’s at a point = 360°)

- 302 - 302 x = 58°

x + 302 = 360

Interior Angles In A Triangle Add To 180°

x 52°

85°

1. Find x

x + 85 + 52 = 180 (’s in a triangle add to 180°)

x + 137 = 180 - 137 - 137

x = 43°

2. Find x

x

46°

x + 90 + 46 = 180

(’s in a triangle add to 180°)x + 136 = 180

- 136 - 136 x = 44°

Base Angles In An Isosceles Triangle Are Equal

1. Find x

x

40°

x + x + 40 = 180 2x + 40 = 180

- 40 - 40 2x = 140

÷ 2 ÷ 2 x = 70°

(base ’s of an isosceles triangle)(’s in a triangle add to 180°)

Exterior Angles Of A Polygon Add To 360°

1. Find x68°

68°

56°

55°

55°

x

x + 68 + 55 + 56 + 68 + 55 = 360 (ext. ’s of a polygon add to 360°)

x + 298 = 360 - 298 - 298

x = 62°

2. Find x in this regular pentagon

Regular means equal sides and angles

x

5x = 360 (ext. ’s of a polygon add to 360°)

÷ 5 ÷ 5 x = 72°

3. A regular polygon has exterior angles of 36°. How many sides does it have?

The number of sides = the number of angles36x = 360

÷ 36 ÷ 36 x = 10

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

divide into triangles from one corner

Interior angle sum = (6 – 2) x 180

n = 6 (or 4 triangles)

(interior angle sum of a polygon)Interior angle sum = 720°

2. Find x

x

Interior angle sum = (5 – 2) x 180

n = 5 or 3 triangles

(interior angle sum of a polygon)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

72°

Exterior angle = 72°x + 72 = 180

x = 108° (adjacent. angles on a straight line = 180°)

Perpendicular Lines- Always cross at right angles.

e.g.

A

B

C D

AB is perpendicular to CD or AB CD

Parallel Lines- Never meet and are always the same distance apart.

e.g.A B

C DF

EAB is parallel to CD or AB ⁄⁄ CD

EF is known as a transversal

Angle Statements and Parallel LinesAlternate 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. e.g. Find x

122°

x

x = 122°(Corresponding angles on parallel lines are equal)

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

77°

x

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

Bearings- Bearings are used to indicate directions- Are measured clockwise from North- Must be expressed using 3 digits (i.e. 000° to 360°)- Compass directions such as NW give directions but are not bearings

e.g. The compass points and their bearings:

E

S

W

NE

SW SE

NW

000°

090°

180°

270°

045°

135°225°

315°e.g. Draw a bearing of 051°:

N

51°

e.g. What is the bearing of R from N?N

37°

Bearing = 180 + 37= 217°

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

6 15

20 4 x y

First calculate ratio between corresponding sides

A

B

C

F

EG

AC = 15EG 6

= 2.5

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

= 10

y = 20 ÷ 2.5

= 8(Similar Triangles) (Similar Triangles)

Parts Of A Circle

Radius Diameter Chord Arc

Sector Segment Circumference Tangent

Angle Properties of CirclesBase Angles Of An Isosceles Triangle Are Equal

e.g.

x40°

Because two sides of the triangle are radii, an isosceles triangle is formed

x = 40° (base ’s of an isosceles triangle)

The Angle At Centre Is Twice The Angle At The Circumference

Proof:

A

A B

x = 180 – 2A

C

x + C = 180180 – 2A + C = 180C = 2A

D D = 2BC + D = 2A + 2BC + D = 2(A + B)

x

e.g. Find x

x

42°x = 2 × 42

x = 84°

( at centre = 2 × at circumf.)

rr

Angle In A Semi Circle Is A Right Angle- This case is a special version of the previous rule

x

x = 90°( in a semi-circle)

Angles On The Same Arc Are Equal

e.g.

Proof:

A

C

B

C = 2AC = 2B

2A = 2BA = B

e.g. Find x:

32°x

x = 32°(’s on the same arc)

There are 2 arcs joining angles

The Angle Between Tangent And Radius Is A Right Angle

e.g.

x

x = 90°(tangent radius)

If Two Tangents Are Drawn From A Point To A Circle They Are The Same Length

e.g.

xy

54° 2x + 54 = 180 - 54 - 54

÷ 2 ÷ 2 x = 63°

2x = 126

( sum isos. triangle)

y + 63 = 90- 63 - 63

y = 27°

(tangent radius)

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.

79°

xx + 79 = 180

- 79 - 79 x = 101°

(opp. ’s, cyc. quad)

Proof:

B 2B A2A

2A + 2B = 360A + B = 180

Exterior Angle Of A Cyclic Quadrilateral Equals Opposite Interior Angle

110°

x

x = 110°

(ext. , cyc. quad)

e.g. Proof:

A

BC

A + B = 180B + C = 180B = 180 – C

A + 180 – C = 180A = C