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Geometry Lines and Angles CONFIDENTIAL 1 Warm Up Find the circumference and area of each circle. 1) 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 C D G H CONFIDENTIAL 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 C D G H CONFIDENTIAL 6 Parallel, perpendicular and skew lines Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG. A B E F C D G H CONFIDENTIAL 7 Identifying types of lines and planes Identify each of the following: A) A pair of parallel segments. KN || PS K L B) A pair of skew segments. LM || RS N M C) A pair of perpendicular segments. P Q MR | RS S R D) A pair of parallel planes. plane KPS || plane LQR. CONFIDENTIAL 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: C D 1a) A pair of parallel segments B E 1b) A pair of skew segments 1c) A pair of perpendicular segments G H 1d) A pair of parallel planes F 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. Same side interior angles CONFIDENTIAL 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 5 transversal: m; Corresponding angles 6 C) Angle 1 and angle 4 m n transversal: l; Alternate exterior angles CONFIDENTIAL 18 Now you try! 3) Identify the transversal and classify the angle pair 2 and 5 in the diagram. 4 l 5 6 m n CONFIDENTIAL 19 Assessment Identify each of the following: B C 1) A pair of perpendicular segments A D 2) A pair of skew segments 3) A pair of parallel segments F G 4) A pair of parallel planes E 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. m 9) Angle 1 and angle 2 n 10) Angle 2 and angle 3 4 11) Angle 2 and angle 4 5 12) Angle 4 and angle 5 3 p CONFIDENTIAL 22 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 C D G H CONFIDENTIAL 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 C D G H CONFIDENTIAL 25 Parallel, perpendicular and skew lines Parallel planes are planes that do not intersect. In the figure, plane ABE || plane CDG. A B E F C D G H 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. Same side interior angles CONFIDENTIAL 31 You did a great job today! CONFIDENTIAL 32