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CHAPTER 1

Mohd Khusaini Majid Mrsm Kota Kinabalu

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1.1 PROPERTIES 1.1 PROPERTIES 1.1 PROPERTIES 1.1 PROPERTIES

OF ANGLES OF ANGLES OF ANGLES OF ANGLES

ASSOCIATED ASSOCIATED ASSOCIATED ASSOCIATED

WITH WITH WITH WITH

TRANSVERSALS TRANSVERSALS TRANSVERSALS TRANSVERSALS

AND PARALLEL AND PARALLEL AND PARALLEL AND PARALLEL

LINESLINESLINESLINES

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Transversal

– a line that intersects two or more parallel lines

Identify transversals, corresponding angles, Identify transversals, corresponding angles, Identify transversals, corresponding angles, Identify transversals, corresponding angles,

alternate angles and interior anglesalternate angles and interior anglesalternate angles and interior anglesalternate angles and interior angles

P

QQQ

S

RPP

PQ and RS are transversals.

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Corresponding angles

e

f

d

c

abTIPS

Corresponding angles

can be identified by

the F shape. The F can

be flipped over or

turned upside down.Shaded angles are

corresponding angles. Angles a

and d are also corresponding

angle.

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Alternate angles

Shaded angles are alternate angles.

Angles a and b also alternate angles.

TIPS

Alternate angles can

be identified by the Z

shape or Z shape.

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Interior angles

Shaded angles are interior angles.

Angles c and d also interior angles.

TIPS

Interior angles can be

identified by the Z

shape or Z shape.

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Corresponding angles are equal

Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines:

a) corresponding angles are equala) corresponding angles are equala) corresponding angles are equala) corresponding angles are equal

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Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines:

b) alternate angles are equalb) alternate angles are equalb) alternate angles are equalb) alternate angles are equal

Alternate angles are equal

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Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines: Determine that for parallel lines:

c) the sum of interior angles is 180c) the sum of interior angles is 180c) the sum of interior angles is 180c) the sum of interior angles is 180°°°°

The sum of interior angles is 180° or interior angles are supplementary

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Exercise 1.1AExercise 1.1AExercise 1.1AExercise 1.1A

1. For each of the following diagrams, list

i. four pairs of corresponding angles.

ii. two pairs of alternate angles.

iii. two pairs of interior angles.

a) b)

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Find the value of x in each of the following.

a) b)

Find the values of corresponding angles, Find the values of corresponding angles, Find the values of corresponding angles, Find the values of corresponding angles,

alternate angles and interior angles, alternate angles and interior angles, alternate angles and interior angles, alternate angles and interior angles,

associated with parallel linesassociated with parallel linesassociated with parallel linesassociated with parallel lines

Example 1Example 1Example 1Example 1

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a) The angles shown in the diagram are

corresponding angles.

Corresponding angles are equal.

Thus, x = 125°.

b) The angles shown in the diagram are interior

angles.

The sum of interior angles is 180°.

Thus, x + 74° = 180°

x = 180° - 74° = 106°

Solution:

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Exercise 1.1BExercise 1.1BExercise 1.1BExercise 1.1B

Find the value of the unknown in each of the

following.

1. 2.

3. 4.

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When two straight lines are intersected by a

transversal, and

Determine if two given lines are parallel Determine if two given lines are parallel Determine if two given lines are parallel Determine if two given lines are parallel

based on the properties of angles associated based on the properties of angles associated based on the properties of angles associated based on the properties of angles associated

with transversalswith transversalswith transversalswith transversals

• if the corresponding angles are equal, then the

two lines are parallel.

• if the alternate angles are equal, then the two

lines are parallel.

• if the interior angles are supplementary, then

the two lines are parallel.

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Example 2Example 2Example 2Example 2

In the diagrams below, KLM, STU and LTV are

straight lines. Determine if the lines KLM and

STU are parallel.

a) b)

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a) The corresponding angles are equal.

Thus, the lines KLM and STU are parallel.

b) If the lines are parallel,

then, LTU = 153°.

Given STU is a straight line.

Hence, STU = 180°.

But LTU + LTS = 153° + 26°

= 179° ≠ 180°

Thus, the lines KLM and STU are not parallel.

Solution:

Corresponding angle

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Exercise 1.1CExercise 1.1CExercise 1.1CExercise 1.1C

1. Determine if the bold lines shown below are

parallel.

a) b)

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2. Determine which two lines are parallel in

each of the following.

a) b)

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The diagram shows a parallelogram ABCD. Given

EDC is a straight line. Calculate all the angles in

the parallelogram.

Solve problems involving the properties of Solve problems involving the properties of Solve problems involving the properties of Solve problems involving the properties of

angles associated with transversalsangles associated with transversalsangles associated with transversalsangles associated with transversals

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DCB = EDA

= 105°

ADC = 180° - DCB

= 180° - 105°

= 75°

DAB = EDA

= 105°

ABC = 180° - DAB

= 180° - 105°

= 75°

Solution:

Alternate angles

Interior angles

Corresponding angles

Interior angle

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Example 4Example 4Example 4Example 4

The diagram shows two parallel lines intersected

by two transversals. Find the values of x and y.

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BSV = ABS

x = 34°

Angle y is the sum of ASW and WSC.

ASW = 180° - UAS

= 180° - 93°

= 87°

WSC = UBS

= 34°

Thus, y = 87° + 34°

= 121°

Solution:

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Exercise 1.1DExercise 1.1DExercise 1.1DExercise 1.1D

1. In the diagram, ABCD is a trapezium and the

line EF is an extension of AB. Given EAD

= 77° and FBC = 25°. Calculate all the

angles in the trapezium.

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2. The following diagrams consist of straight

lines. Find the values of x and y.

a) b)

c)

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Summary Summary Summary Summary LINES AND ANGLES IILINES AND ANGLES IILINES AND ANGLES IILINES AND ANGLES II

Parallel lines

* The lines that will never meet

* The lines that are always the same

distance apart.

Transversal

* A line that intersects two or

more parallel lines.

Transversal and parallel lines

Corresponding angles

* are equal

* can be identified by the

F shape.

Corresponding angles

Alternate angles

* are equal

* can be identified

by the Z shape.

Interior angles

* have a sum of 180°

* can be identified

by the UUUU shape.

Interior anglesAlternate angles

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Reference

Yoong Kwee Soon, Sim Kwang Yaw, Ding Hong

Eng, Noorliah Binti Ahmat, Lai Jun Siew,

Mathematics FORM 3, Ebiza Sdn. Bhd., 2004.

Prepared by: mohd khusaini majid Mrsm Kota Kinabalu