Transcript Slide 1
Geometry
Lines and Angles
CONFIDENTIAL 1
1)
Warm Up
Find the circumference and area of each circle.
2) 80 cm 3.8 m CONFIDENTIAL 2
Parallel, 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 3
Parallel, 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 E F C D G H CONFIDENTIAL 4
Parallel, 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 E F G C H CONFIDENTIAL D 5
Parallel, perpendicular and skew lines
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.
A B E F G C H CONFIDENTIAL D 6
Parallel, perpendicular and skew lines
Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG.
E A C G F H B D CONFIDENTIAL 7
Identifying types of lines and planes
Identify each of the following: A) A pair of parallel segments.
KN || PS K B) A pair of skew segments.
LM || RS N C) A pair of perpendicular segments.
MR | RS S D) A pair of parallel planes.
plane KPS || plane LQR.
P CONFIDENTIAL M R L Q 8
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 9
Now you try!
Identify each of the following: 1a) A pair of parallel segments 1b) A pair of skew segments 1c) A pair of perpendicular segments 1d) A pair of parallel planes B F C G E D H J CONFIDENTIAL 10
Angle 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.
exterior angles interior angles exterior angles CONFIDENTIAL 11
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 12
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 13
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 14
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. CONFIDENTIAL Same side interior angles 15
Classifying pairs of angles
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 16
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 CONFIDENTIAL 17
Identifying angle pairs and transversals
Identify the transversal and classify each angle pair.
A) Angle 1 and angle 5 transversal: n; Alternate interior angles 4
l
B) Angle 3 and angle 6 transversal: m; Corresponding angles 6 5 C) Angle 1 and angle 4 transversal: l; Alternate exterior angles
m n
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Now you try!
3) Identify the transversal and classify the angle pair 2 and 5 in the diagram.
m
6 5
n
4
l
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Assessment
Identify each of the following: 1) A pair of perpendicular segments 2) A pair of skew segments 3) A pair of parallel segments 4) A pair of parallel planes A E B F D C G H CONFIDENTIAL 20
Give an example of each angle pair.
5) Alternate interior angles 6) Alternate exterior angles 7) Corresponding angles 8) Same side interior angles CONFIDENTIAL 21
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 4 5 3
p m n
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Let’s review
Parallel, 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 E F C D G H CONFIDENTIAL 23
Parallel, 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 E F G C H CONFIDENTIAL D 24
Parallel, perpendicular and skew lines
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.
A B E F G C H CONFIDENTIAL D 25
Parallel, perpendicular and skew lines
Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG.
E A C G F H B D CONFIDENTIAL 26
Angle 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.
exterior angles interior angles exterior angles CONFIDENTIAL 27
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 28
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 29
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 30
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. CONFIDENTIAL Same side interior angles 31
You did a great job today!
CONFIDENTIAL 32