Lines, angles and polygons

Labelling Lines

A straight line can be considered to have

infinite length and no measurable width.

15.14cm

A

B

A line segment is of finite length, e.g line segment AB

has end-points A and B.

Labelling Angles

When two line segments meet at one point, the angle formed is the measure of rotation of one of the line segments to the other.

Angle DÊF =35°

D

E

F

Labelling Triangles

Use capital letters for vertices and the corresponding lower-case letters for each opposite side, the triangle then being described as ∆ABC

The corner of a triangle is called a vertex

The plural of vertex is vertices

A

B

C

a

b

c

Equal sides and Parallel sides

B

C

AB = AC

A

B

C

D

AB is parallel to DC, or AB//DC.

AD is parallel to BC, or AD//BC.

A

Isosceles triangle

Angles in Triangles

E

D

F

G

Interior angle

Exterior angle

Complementary angles add up to 90°

Supplementary angles add up to 180°

Parallel and Perpendicular Lines

Parallel lines are always the same distance apart:

Perpendicular lines are lines which intersect at right angles.

Equidistant

Intersect – cross each other

Relationships between lines in the plane

A

B

C

D

E

F

A+B+C=180

A=D

B=E

C=F

Three lines that intersect in one point make 6 angles

Two pairs of parallel lines, forming a parallelogram.

Corresponding angles and alternate angles

A

A

Alternate Angles

C

C

Corresponding Angles

Use alternate angles to prove that opposite angles of a parallelogram are equal:

A

B

A=B

p=q

r=s

p

q

r

s

∴ r+p = q+s

Alternate angles

are equal

Corresponding angles

are equal

Angles on a Line

B

B

A

A

X

Y

X+Y=180°

Vertically opposite angles are equal

Angles on a straight line

Angles add up to 180°.

All angles add up to 360°

A+B+A+B = 360°

Three interior angles of a triangle add up to 180°

A

B

C

D

E

F

(A+B+C) = 180°

Use the angle sum of a triangle to prove that the angle sum of a quadrilateral is 360°.

(D+E+F) = 180°

A+B+C+D+E+F = 360°

Prove the exterior angle of a triangle k = sum of two opposite Interior angles, i and j

Y

X

A

B

i

j

k

X = Y (Alternate angles)

A=B (Corresponding angles)

A+X =Y+B

Y+B = k

A+X = i+J

i and j are interior and

opposite each other

k is exterior angle

Angles marked by letters

X

y

46°

136°

76°

Calculate the angles marked

x and y.

A

B

C

D

E

Triangle BCD is isosceles

a

b

a = b = (180°-136°)/2=22°

76+x+a = 180

46+y+b = 180

76+x+22=180

98+x=180

x=180-98

x= 82°

68+y = 180

y =112°

Finding angles in a polygon

A polygon with n sides can be split into n – 2 triangles.

Polygon

A shape with many sides

How many sides on this polygon?

6

1

2

3

4

5

6

How many triangles can I make?

4

1

2

3

4

The number of triangles is always 2

less than the number of sides

Each triangle has an angle sum of 180°.

The interior angle sum is (n – 2) x 180°

So a 4 sided shape, a quadrilateral, has

(4-2) x 180° = 360°

A 5 sided shape, a pentagon, has

(5-2) x 180 = 540°

Vertexes or vertices

At each vertex, the sum of the interior and exterior angles is 180°.

The vertex is the place

where two sides meet,

a corner of the shape

Vertex

x

y

X+Y = 180°

For n vertices, the sum of n interior and n exterior angles is

But the sum of the interior angles is (n – 2) x 180°,

the sum of the exterior angles is always 2 x 180° = 360°

∴

n x 180°.

2003-10-10T15:08:00