Angle Construction

Angles An angle is formed when two rays meet at a point called the vertex. Angles are usually measured in degrees using a protractor. Angle measures range from 0° to 360°. Angles are classified according to their size in degrees. Acute Angle Right Angle Obtuse Angle Straight Angle Reflex Angle measure is between 0° and 90° measure is 90° measure is between 90° and 180° measure is 180° measure is between 180° and 360°

Examples: 1. Identify the following angles as acute, right, obtuse, straight, or reflex.

Identify the type of angle:  68° b) 215° c) 91° d) 32° e) 180° f) 99° g) 193° h) 265°

Complementary Angles: two angles that have measures (size) that add up to 90° Supplementary Angles: two angles that have measures that add up to 180° Opposite Angles: two angles that are opposite each other have the same measure Congruent Angles: Two angles with the same measure are referred to as congruent. For example, opposite angles are congruent.

  Examples: Given each of the following angles, determine the size of the complement and/or the size of the supplement (if they exist).   75° 43° 103° 87°  300° Sort the following angles into pairs of complementary and supplementary : 1 = 42° 2 = 107° 3 = 59° 4 = 48° 5 = 121° 6 = 31° 7 = 19° 8 = 73°

Assignment: Angles Curriculum Outcomes: 10E2.AC.1. demonstrate an understanding of angles, including acute, right, obtuse, straight, and reflex, by: drawing, replicating and constructing, bisecting, and solving problems   Assignment: Angles

Angle Bisectors and Perpendicular Lines To bisect something is to cut it into two equal parts. An angle is bisected by a ray that divides it into two angles of equal measure. The ray that divides the angle is called an angle bisector. Perpendicular lines are two lines that form a right angle. Examples: 1.

2. Estimate the size of the angle shown below and then bisect it.

3. By inspections, determine which of the following lines are perpendicular.. 4. The size of an angle is the same as the supplement of the bisected angle, what is the angle?

Assignment: Angle Bisectors and Perpendicular Lines Curriculum Outcomes: 10E2.aC.1. demonstrate an understanding of angles, including acute, right, obtuse, straight, and reflex, by drawing, replicating and constructing, bisecting, and solving problems   Assignment: Angle Bisectors and Perpendicular Lines

Parallel Lines and Transversals Definitions: transversal: a line that intersects two or more lines corresponding angles: two angles that occupy the same relative position at two different intersections opposite angles: angles created by intersecting lines that share only a vertex interior angles: on the same side of the transversal: these angles are supplementary (different intersections) exterior angles: on the same side of the transversal: these angles are supplementary (different intersections) alternate interior angles: angles in opposite positions between two lines intersected by a transversal and also on alternate sides of the same transversal (different intersections) alternate exterior angles: angles in opposite positions outside two lines intersected by a transversal (different intersections)

Parallel Lines and Transversals If two parallel lines are intersected by a transversal: • the alternate interior (exterior) angles are equal; • the corresponding angles are equal; • the interior (exterior) angles on the same side of the transversal are supplementary. If you are given two lines cut by a transversal: • alternate interior (exterior) angles are equal; OR • corresponding angles are equal; OR • interior (exterior) angles on the same side of the transversal are supplementary; then you can conclude that the lines are parallel.

Examples:   Consider the diagram below, in which ℓ1 is parallel to ℓ2. What are the measures of the three indicated angles? Explain how you reached your answers.

2. Given the diagram below, identify all the pairs of parallel lines and explain your selection.

Assignment: Parallel Lines and Transversals Curriculum Outcomes: 10E2.aC.2. solve problems that involve parallel, perpendicular, and transversal lines, and pairs of angles formed between them. Assignment: Parallel Lines and Transversals

Non – Parallel Lines and Transversals   Important Note: Non – parallel lines follow the same definitions as parallel lines with the exception that we cannot determine the size of the angles just their positioning.

• ∠3 and ∠5, ∠4 and ∠6 are pairs of alternate interior angles. Example: Consider the diagram below: t is a transversal that intersects ℓ1 and ℓ2. • ∠1 and ∠5, ∠4 and ∠8, ∠2 and ∠6, ∠3 and ∠7 are pairs of corresponding angles. • ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8 are pairs of opposite angles. • ∠3 and ∠5, ∠4 and ∠6 are pairs of alternate interior angles. • ∠2 and ∠8, ∠1 and ∠7 are pairs of alternate exterior angles. • ∠3 and ∠6, ∠4 and ∠5 are pairs of interior angles on the same side of the transversal. • ∠1 and ∠8, ∠2 and ∠7 are pairs of exterior angles on the same side of the transversal.

Identify each of the following and specify which line and transversals you are using. a) an interior angle on the same side of the transversal as ∠6 b) an angle corresponding to ∠2 c) an angle corresponding to ∠4 d) an alternate interior angle to ∠4

In the diagram below, measure and record the sizes of the angles In the diagram below, measure and record the sizes of the angles. Identify pairs of equal angles and state why they are equal.

Assignment: Non – Parallel Lines and Transversals Curriculum Outcomes: 10E2.aC.2. solve problems that involve parallel, perpendicular, and transversal lines, and pairs of angles formed between them. Assignment: Non – Parallel Lines and Transversals

Spatial Geometry Rotations:   Rotations: a rotation is a transformation in which a figure is turned or rotated to rotate a figure, you need to know the amount and direction of the rotation as well as the point of rotation (called the turn centre) we will look at 90°, 180°, and 270° rotations