CONFIDENTIAL 1

GeometryGeometry

Lines and AnglesLines and Angles

CONFIDENTIAL 2

Warm UpWarm Up

Find the circumference and area of each circle.

1) 2)

80 cm 3.8 m

CONFIDENTIAL 3

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Pairs of lines can relate to each other in four different ways: intersecting lines, parallel lines, perpendicular lines and skew lines. These concepts are useful for

understanding and solving various geometry problems.

CONFIDENTIAL 4

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Parallel lines (||) are lines that never intersect i.e. they are coplanar. The distance between the two lines is

fixed and the two lines go in the same direction. In the figure, AB || EF and EG || FH.

A B

C D

E F

G H

CONFIDENTIAL 5

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Perpendicular lines (|)are lines that intersect at one point and form a 90° angle i.e. two different straight lines on the

same plane in two different directions who meet each other at only right angles are called Perpendicular Lines.

In the figure, AB | AE and EG | GH.

A B

C D

E F

G H

CONFIDENTIAL 6

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

A B

C D

E F

G H

Skew lines are not coplanar. Skew lines only happen in space. Skew lines never intersect because they are not on the same plane. Skew lines are difficult to draw because

they exist in the three dimensional space. In the figure, AB and EG are skew.

CONFIDENTIAL 7

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG.

A B

C D

E F

G H

CONFIDENTIAL 8

Identifying types of lines and planesIdentifying types of lines and planes

K L

P Q

N M

S R

Identify each of the following:

A) A pair of parallel segments. KN || PS

B) A pair of skew segments. LM || RS

C) A pair of perpendicular segments. MR | RS

D) A pair of parallel planes. plane KPS || plane LQR.

CONFIDENTIAL 9

Referring to the figure, we can conclude:

AB is perpendicular to CL and  CIB is 90° •KE intersects IB at point J •GH is parallel to AB •KD is perpendicular to MH and  KLH is 90° •IL intersects JM at point K •EF intersects GL at point M, intersects IL at point K and IB at point J

CONFIDENTIAL 10

1a) A pair of parallel segments

1b) A pair of skew segments

1c) A pair of perpendicular segments

1d) A pair of parallel planes

Now you try!

Identify each of the following:

C D

G H

B E

F J

CONFIDENTIAL 11

Angle pairs formed by a transversalAngle pairs formed by a transversal

A transversal is a line that intersects two coplanar lines at two different points. The traversal t and the other two lines

r and s form eight angles.

interior angles

exterior angles

exterior angles

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Corresponding angles are created where a transversal crosses other (usually parallel) lines. The corresponding

angles are the ones at the same location at each intersection i.e. angle 1 and angle 5.

Corresponding angles

CONFIDENTIAL 13

Alternate interior angles are created where a transversal crosses two (usually parallel) lines. Each pair of these

angles are inside the parallel lines, and on opposite sides of the transversal i.e. angle 3 and angle 5.

Alternate interior angles

CONFIDENTIAL 14

Alternate exterior angles are created where a transversal crosses two (usually parallel) lines. Each pair of these

angles are outside the parallel lines, and on opposite sides of the transversal i.e. angle 1 and angle 7.

Alternate exterior angles

CONFIDENTIAL 15

Same side interior angles crosses two (usually parallel) lines. Each pair of interior angles are inside

the parallel lines, and on the same side of the transversal. i.e. angle 3 and angle 6.

Same side interior angles

CONFIDENTIAL 16

Classifying pairs of Classifying pairs of anglesangles

4 3

1 2

8 7

5 6

Give an example of each angle pair.

A) Corresponding angles Angle 4 and angle 8

B) Alternate interior angles Angle 4 and angle 6

C) Alternate exterior angles Angle 2 and angle 8

D) Same side interior angles Angle 4 and angle 5

CONFIDENTIAL 17

Now you try!

Give an example of each angle pair.

2a) Corresponding angles

2b) Alternate interior angles 2c) Alternate exterior angles

2d) Same side interior angles

1 2 3 45 6 7 8

CONFIDENTIAL 18

Identifying angle pairs and transversalsIdentifying angle pairs and transversals

Identify the transversal and classify each angle pair.

A) Angle 1 and angle 5 transversal: n; Alternate interior angles

B) Angle 3 and angle 6 transversal: m; Corresponding angles

C) Angle 1 and angle 4 transversal: l; Alternate exterior angles

1 2 3

4

56

m

l

n

CONFIDENTIAL 19

1 2 3

4

56

m

l

Now you try!

3) Identify the transversal and classify the angle pair 2 and 5 in the diagram.

n

CONFIDENTIAL 20

Assessment

1 )A pair of perpendicular segments

2 )A pair of skew segments

3 )A pair of parallel segments

4 )A pair of parallel planes

Identify each of the following:

B C

F G

A D

E H

CONFIDENTIAL 21

Give an example of each angle pair.

5) Alternate interior angles 6) Alternate exterior angles

7) Corresponding angles

8) Same side interior angles

6 2 3 75 1 4 8

CONFIDENTIAL 22

Identify the transversal and classify each angle pair.

9) Angle 1 and angle 2 10) Angle 2 and angle 3

11) Angle 2 and angle 4

12) Angle 4 and angle 5

1 2

3

5

4

m

n

p

CONFIDENTIAL 23

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Parallel lines (||) are lines that never intersect i.e. they are coplanar. The distance between the two lines is

fixed and the two lines go in the same direction. In the figure, AB || EF and EG || FH.

A B

C D

E F

G H

Let’s review

CONFIDENTIAL 24

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Perpendicular lines (|)are lines that intersect at one point and form a 90° angle i.e. two different straight lines on the

same plane in two different directions who meet each other at only right angles are called Perpendicular Lines.

In the figure, AB | AE and EG | GH.

A B

C D

E F

G H

CONFIDENTIAL 25

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

A B

C D

E F

G H

Skew lines are not coplanar. Skew lines only happen in space. Skew lines never intersect because they are not on the same plane. Skew lines are difficult to draw because

they exist in the three dimensional space. In the figure, AB and EG are skew.

CONFIDENTIAL 26

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG.

A B

C D

E F

G H

CONFIDENTIAL 27

Angle pairs formed by a transversalAngle pairs formed by a transversal

A transversal is a line that intersects two coplanar lines at two different points. The traversal t and the other two lines

r and s form eight angles.

interior angles

exterior angles

exterior angles

CONFIDENTIAL 28

Corresponding angles are created where a transversal crosses other (usually parallel) lines. The corresponding

angles are the ones at the same location at each intersection i.e. angle 1 and angle 5.

Corresponding angles

CONFIDENTIAL 29

Alternate interior angles are created where a transversal crosses two (usually parallel) lines. Each pair of these

angles are inside the parallel lines, and on opposite sides of the transversal i.e. angle 3 and angle 5.

Alternate interior angles

CONFIDENTIAL 30

Alternate exterior angles are created where a transversal crosses two (usually parallel) lines. Each pair of these

angles are outside the parallel lines, and on opposite sides of the transversal i.e. angle 1 and angle 7.

Alternate exterior angles

CONFIDENTIAL 31

Same side interior angles crosses two (usually parallel) lines. Each pair of interior angles are inside

the parallel lines, and on the same side of the transversal. i.e. angle 3 and angle 6.

Same side interior angles

CONFIDENTIAL 32

You did a great job You did a great job today!today!