Angles, Lines and Triangles resources... · Outcomes for Lesson 1 In this lesson we will review: • Angle terminology • Classification of angles according to size • Adjacent,

Angles, Lines

and Triangles

Grade 10 CAPS Mathematics

Video Series

Outcomes for this Video

In this DVD you will:

• Revise factorization.

LESSON 1.

• Revise simplification of algebraic fractions.

LESSON 2.

• Discuss when trinomials can be factorized.

LESSON 3.

2

In this video the focus will be on:

Basic results regarding lines and angles

(Lesson 1)

Basic results regarding triangles

(Lesson 2)

Congruency of triangles

(Lesson 3

Basic results about

angles and lines


Video Series

Lesson 1

Outcomes for Lesson 1

In this lesson we will review:

• Angle terminology

• Classification of angles according to size

• Adjacent, complementary and supplementary angles

• Vertical opposite angles

• Perpendicular and parallel line segments

• Corresponding, alternate and co-interior angles

Angles: Some definitions and terminology

Can view an angle as the union of two rays

(line segments) which have a common endpoint.

ˆ ˆ Alternative notations: or or or .

The common endpoint is known as the vertex and

the rays (segments) the sides (arms) of the angle.

RPQ RPQ P P

To avoid confusion

we lable angles.

Measure angles in degrees written as .

Examples: 70 and 285 QPR ABC

Angles are measured

with a .protractor

Classification of angles according to size

is an angle

0 90

acute is a angle

90

right is an angle

90 180

obtuse

is a angle

180

straight is a angle

180 360

reflex is a

360

revolution

Adjacent, Complementary and Supplementary angles

and are angles

They have a common vertex

They have a common side

They are on opposite sides of common side

B

BD

adjacent

and are angles

90

complementary

and are angles

180

supplementary

and can also be classified as

angles

adjacent complementary

and can also be classified as

angles

adjacent supplementary

Vertically opposite angles

180 180

Assume that and intersect at .AB CD E

180 180

Both equal to 180

In a similar way: Both equal to 180

and are vertically opposite angles

and are also vertically opposite angles

If two lines intersect then vertically opposite angles are equal

Conclusion :

Perpendicular line segments

if one of the four

angles at is a right angle.

AB CD

E

More examples:

Why are the other three angles

in figure above also right angles?

Challenge :

Interior, Exterior, Corresponding and Alternate angles

Given: and and transversal cutting both line segments.AB CD EF

Angles between and ,

ˆ ˆ ˆˆ i.e. 3, 4, 5 and 6.

AB CD

Interior angles :

Angles outside and ,

ˆ ˆ ˆˆ i.e. 1, 2, 7 and 8.

AB CD

Exterior angles :

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ 1 & 5 ; 3 & 7 ; 2 & 6 ; 4 & 8

Pairs of corresponding angles :

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ 1 & 8 ; 2 & 7 ; 3 & 6 ; 4 & 5

Pairs of alternate angles :

ˆ ˆ ˆ ˆ 4 & 6 ; 3 & 5Pairs of Co - Interior angles :

Angle relationships when lines are parallel

Given: and transversal cutting both line segments.AB CD EF

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ 1 5 ; 3 7 ; 2 6 ; 4 8

Corresponding angles equal :

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ 1 8 ; 2 7 ; 3 6 ; 4 5

Alternate angles equal :

ˆ ˆ ˆ ˆ 4 6 180 ; 3 5 180

Co - interior angles supplementary :

Two line segments cut by a transversal will be parallel

Corresponding angles are equal

Alternate angles are equal

Co-interior angles are supplementary

Conclusion :

Tutorial 1: Part 1: Basics on Angles and Lines

PAUSE VIDEO

• Do Tutorial 1 Part 1

• Then View Solutions

1 Calculate the value of .x

2 Calculate the values of and .x y

3 50 and 80 .

If ,

show that .

A ACB

ACD DCE x

BA CD

Tutorial 1: Part 1: Problems 1 to 3: Suggested Solutions

1 Calculate the value of .x

2 Calculate the values of and .x y

3 50 and 80 .

If ,show that .

A ACB

ACD DCE x BA CD

2 3 180 Straight angle

6 180 30

x x x

x x

132 180 Straight angle 48

2 30 96 30 66

x x

x

2 30 180 Straight angle

66 48 180 180 114 66

y x x

y y

180 80

2 100 50

BCE x x

x x

50

Alternate angles are equal

BAC ACD

Alt BA CD BAC ACD

4 In the figure .

ˆˆ 40 and 30 .

Determine with reasons:

4.1

4.2

4.3

4.4

4.5 What is the relationship between

the three angles of ?

4.

AB CD

B A

ACD

ECD

ACB

ACE

ABC

6 What is the relationship between

, and ?A B ACE

PAUSE VIDEO

• Do Tutorial 1: Part 2


5 and .

Show that .

BA ED BC EF

ABC DEF

Tutorial 1: Part 2: Basics on Angles and Lines

4 In the figure .

ˆˆ 40 and 30 .

Determine with reasons:

4.1

4.2

4.3

4.4

4.5 What is the relationship between

the three angles of ?

4.

AB CD

B A

ACD

ECD

ACB

ACE

ABC

6 What is the relationship between

, and ?A B ACE

Tutorial 1: Part 2: Problem 4: Suggested Solution

4.1 Alt 30ACD BAC

4.2 Cor 40ECD ABC

4.3 180 Straight Angle

180

180 30 40 110

BCE

ACB ACD ECD

ACB

4.4

30 40 70

ACE ACD ECD

4.5

30 40 110 180

Sum of three angles is 180

BAC ABC ACB

Exterior angle of the triangle is equal to the sum of two opposite interior angles

4.6 From 4.1 & 4.2ACE ACD ECD BAC ABC

Tutorial 1: Part 2: Problems 5: Suggested Solution

5 and .

Show that .

BA ED BC EF

ABC DEF

Assume that BC DE G

GCorresonding angles equal

Now

ABC

AB DEDGC

Corresonding angles equal

BC EFDEF

Basic Results on

Triangles


Video Series

Lesson 2



• Basic triangle notations and terminology

• Classification of triangles according to side lengths

• Classification of triangles according to angle sizes

• Proof that exterior angle for any triangle is equal to

the sum of the two opposite interior angles

• Proof that the sum of the interior angles for any

triangle is equal to

• Some techniques linked to proofs in Geometry

180

Basic Triangle Notations and Terminologies

is an for .

is the (height) for if base is .

is one of three possible for .

AF A

BG ABC AC

HC ABC

angle bisector

altitude

medians

, and are the sides of .

, and are the vertices of .

, and are the angles of .

AB c AC b BC a ABC

A B C ABC

BAC A ABC B ACB C ABC

Classification of Triangles according to side lengths

Triangle has sides of different lengths.

All three angle measures are different.

Scalene triangle

Know :

Triangle has two sides of the same length.

D F

Isosceles triangle

Know :

Triangle has three sides of equal length.

60G H I

Equilateral triangle

Know :

Triangle where angle contained within

two equal sides is a right angle.

45K L

Isosceles right - angled triangle

Know :

Classification of Triangles according to angle sizes

Triangle has three acute angles.

Acute - angled triangle

Triangle has one obtuse angle and two acute angles.

Obtuse - angled triangle

Triangle has a right angle for one of its angles.

Side opposite the right angle, is called the .

The other two sides contain the right angle.

Right - angled triangle

hypothenuse

Right-angled triangle in which the

other two angles are both equal to 45 .

Right - angled isosceles triangle

Exterior angle of a is equal to sum of interior opposite angles

ˆ ˆ1 corr B CE BA

Proof :

Any with

ˆ ˆ Prove that

Draw

ABC D BC

ACD A B

CE BA

Formal Proof

Given :

Required :

Construction :

ˆ ˆ 2 3

ˆ ˆ 1 3

ˆ ˆ 1 2

BAE

CBD

ACF

Three possibilities ˆ ˆ2 alt A CE BA

ˆ ˆ ˆ ˆ1 2

ˆ ˆ

A B

ACD A B

Sum of the interior angles of any triangle is equal to 180

ˆ ˆ ˆAssume that 1; 2 and 3P QPR MPQ RPN

Proof :

Any

Prove that 180

Draw with

PQR

P Q R

MN QR P MN

Formal Proof

Given :

Required :

Construction :

ˆ ˆalt 2 and alt 3 Q MN QR R MN QR

ˆ ˆ ˆ1 2 3

180 is a straight angle

P Q R

MPN

MPN

Example 1: Application - sum of interior angles of a triangle

s180 Sum of 3 of BCG B BGC BCG

Proof :

In , 90 .

is any point on .

.

Prove that

ABD D

C BD

CG AB

BCG A

90 90B BGC

sSimilarly: 180 Sum of 3 of

90 90

A B D ABD

B D

Both equal to 90BCG A B

Example 2: Application - exterior and opposite interior angles of a triangle

In , is drawn with .

Given that 2 .

Prove that .

PQR QS R QS

PRS P

P Q

and

2 GivenPRS P

2 Both equal to P Q P PRS

2Q P P P

Ext angle of PRS P Q PQR

Proof :

Tutorial 2: Basic Results on Triangles

1 In , is drawn with .

Given that:

and

bisects .

Prove that .

ABC BD A BD

B C

AE DAC

AE BC

PAUSE VIDEO

• Do Tutorial 2


2 In the figure:

and are the bisectors of

and respectively.

Prove that 90 .

MB ND

MO NO

NMB MND

MON

Tutorial 2: Problem 1: Suggested Solution

1 In , is drawn with .

Given that:

and

bisects .

Prove that .

ABC BD A BD

B C

AE DAC

AE BC

Ext. angle of DAC B C ABC

Proof :

2 2 and

DAE EAC xx y y y x y

B C y

Alt

or Corr

EAC ACB AE BC

DAE ABC AE BC


2 In the figure:

and are the bisectors of

and respectively.

Prove that 90 .

MB ND

MO NO

NMB MND

MON

Co-interior angles and 180 MB NDBMN MND

Proof :

bisects and bisects 2 2 180 MO BMN NO MNDNMO MNO

90NMO MNO

sSum of 3 of But 180 MNOMON NMO MNO

180 90 90

Congruency

of triangles


Video Series

Lesson 3



• Four cases of congruency of triangles

• Some proofs linked to isosceles triangles

• The Theorem of Pythagoras

Four cases of congruency of triangles

Congruent triangles are triangles which have same and :

1 corresponding sides equal and 2 corresponding angles equal.

Definition : shape area

s

If two sides and the included angle are equal.

In and

1

2

3

, ,

ABC DEF

AB DE

BC EF

ABC DEF

ABC DEF S S

Case 1 :

s

If three sides are equal.

In and

1

2

3

, ,

GHI KLM

GH KL

GI KM

HI LM

GHI KLM S S S

Case 2 :

s

If two angles and corresponding side are equal.

In and

1

2

3

, ,

OPQ RST

OPQ RST

PQO STR

OP RS

OPQ RST S

Case 3 :

s

If a right-angle, hyphothenuse and one side are equal.

In and

1

2

3

hyp, ,90

UVW XYZ

UV XY

VW YZ

UWV XZZ

UVW XYZ S

Case 4 :

Cases where two triangles are not necessarily congruent

If three angles of one triangle

are eqaul to corresponding three

angles of another triangle.

Case 1 :

Different shapes

Different areas

ABC EFD

Same shape

Different areas

ABC DEF

If two sides and non-included angle of one triangle are

equal to corresponding two sides and a non-included

angle of another triangle.

Case 2 :

Tutorial 3: Congruent Triangles

s1 For each pair of state with reasons if they are congruent or not.

Triangles are not necessarily drawn accurately.

Note :

PAUSE VIDEO

• Do Tutorial 3


2 State with reasons whether the two triangles in each figure

below are congruent.

Name triangles with their vertices in the correct order.


s1 For each pair of in the figures below, state if they are

congruent or not. State your reasons.

Triangles are not necessarily drawn accurately.

Note :

Position 2

Position 3

But position 1 & 3 is

corresponding to position 2 & 3

ABC XYZ

B Y

C Z

AC

YZ

Indicated by //

Indicated by ///

Same lable

, ,

FD LK

FE LM

F L

FED LMK S S

Note:

Positions 1 & 3

Positions 1 & 2

FD LK

FE LM


2 State with reasons whether the two triangles in each figure

below are congruent.

Name triangles with their vertices in the correct order.

In and

Indicated by ///

Vertically opposite

and Alternate angles

But & positions 1 & 3 and

& positions 3 & 1 are not given as equal.

s AEB CED

AE CE

AEB CED

A D B C

A C

B D

AED

CED

In and

Indicated by /

Indicated by //

Common

, ,

sPNR PQR

PN PQ

NR QR

PR PR

PNR PQR S S S

The angles at the base of an isosceles triangle are equal

Any isosceles with

Prove that

Bisect with so that

ABC AB AC

B A

BAC AD D BC

Given :

Required :

Construction :

In and

Given

Construction

Common

, ,

Position 2

s ABD ACD

AB AC

BAD CAD

AD AD

ABD ACD S S

B C

Proof :

If two angles of a triangle are equal, the triangle is isosceles

Any with

Prove that

Bisect with with

ABC B C

AB AC

BAC AD D BC

Given :

Required :

Construction :

In and

Given

Construction

Common

, ,

Positions 1 & 2

s ABD ACD

B C

BAD CAD

AD AD

ABD ACD S

AB AC

Proof :

Informal Investigation: Theorem of Pythagoras

2

2

Cut up square with area into 4 pieces as indicate.

These 4 pieces together with square of area

fit precisely into square on the hypothenuse.

a

b

2 2 2 with 90ABC C c a b

Theorem of Pythagoras :

Formal Proof: Theorem of Pythagoras

In a right-angled triangle the square of the

hypothenuse is equal to the sum of the squares of the other two sides.

Theorem of Pythagoras :

Draw and DM BA EF CHProof :

, ,BAC EFB S AC BF b

Similarly: each with area of .2

bcBAC CHD DME EFB

2 2 2

with 90

Prove that

Complete the square on

ABC BAC

a b c

EBCD BC

Given :

Required :

Construction :

2 2area of is and area of is FA c b MFAH c b EBCD a

22 2 2 2 2Now 4 2 2

2

bca c b c bc b bc c b

Converse Theorem of Pythagoras

If the square of one side of

a triangle is equal to the sum of the squares of the other two sides,

then the angle contained by these sides is a right angle.

Converse Theorem of Pythagoras :

2 2 2

2 2

2

90 Pythagoras

Construction

Given

E e d f

a c

b

b e

Proof :

s

Construction

In and : Construction

Proved

c f

ABC DEF a d

b e

2 2 2 with

Prove 90

Construct with 90 , and .

ABC b a c

B

DEF E f c d a

Given :

Required :

Construction :

, ,

90

ABC DEF S S S

B E

Pythagorean Triples

2 2 2

Is a set of three nonzero whole numbers , and

represented as a triple , , such that .

a b c

a b c a b c

Pythagorian Triple :

3,4,5 ; 6,8,10 ;

5,12,13 ; 10;24;26 ;

Examples :

There are infinitely many Pythagorean triples

3,4,5 is a Pythagorean triple

i.e. have only 1 as a common factor

primitive

Application of Theorem of Pythagoras

2 2 22 4 Pythagoras

4 16 20 2 5

a

a

2 2 2

22 2

2 Pythagoras

2 2 5

4 20 24 2 6

b a

b

b

2 2 2

22 2

3 Pythagoras

3 2 6

9 24 33

c b

c

c

Use the Theorem of Pythagoras to find the

missing lenghts in the given diagram.

Tutorial 4: Isosceles triangles and Pythagoras

1 In , 30 , 45 and .

is a point on such that .

1.1 Calculate and .

1.2 Prove by calculating angles that is equilateral.

ABC A C BD AC

E AB AE ED

ABC ABD

DEB

PAUSE VIDEO

• Do Tutorial 4


2 2 2 2

2 The diagonals of a quadrilateral

cut each other at right angles at .

Prove that .

ABCD E

AB CD AD BC

3 Use the Theorem of Pythagoras to find the



s1.1 180 Sum of 3 of

180 30 45 180 75 105

ABC A B ABC

1 In , 30 , 45 and .

is a point on such that .

1.1 Calculate and .

1.2 Prove by calculating angles

that is equilateral.

ABC A C BD AC

E AB AE ED

ABC ABD

DEB

is a right-angled : 90 90 30 60BDA ABD A

1.2 Ext. of 30 30 60 BED EAD EDA EDB EA ED

sBut 180 Sum of 3 of

180 60 60 60

EDB EDB BED

In all three angles

have a measure of 60 .

DEB

is equilateral.EB ED BD DEB

2 2 2 2 2 2

2 2 2 2

2 2

+

= re-grouped

Pythagoras

AB CD AE EB EC ED

AE ED EC EB

AD BC


2 2 2 2

2 The diagonals of a quadrilateral cut each at

right angles at . Prove that .

ABCD

E AB CD AD BC

2 2 2

2 2 2

Pythagoras

Pythagoras

AB AE EB

CD EC ED


3 Use the Theorem of Pythagoras to find the


2 2 22 3 Pythagoras

4 9 13

x

x

2

2 217 3 Pythagoras

17 9 8

y

8 2 2y

REMEMBER!

•Consult text-books for additional examples.

•Attempt as many as possible other similar

examples on your own.

•Compare your methods with those that were

discussed in the Video.

•Repeat this procedure until you are confident.

•Do not forget:

Practice makes perfect!

End of Video on Angles,

Lines and Triangles

Documents

Angles, Lines and Triangles resources... · Outcomes for Lesson 1 In this lesson we will review: • Angle terminology • Classification of angles according to size • Adjacent,