Course 3 Points, Lines, Planes, and Angles The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°. F K J G H
Course 3 Points, Lines, Planes, and Angles A right angle can be labeled with a small box at the vertex. Reading Math A right angle measures 90°.
Course 3 Points, Lines, Planes, and Angles Complementary angles: Angles whose measures sum to 90°. A right angle measures 90 °. Angle symbol ∡ Supplementary angles: Angles whose measures sum to 180°. A straight line measures 180 °.
Course 3 Points, Lines, Planes, and Angles Vertical Angles: Angles formed by 2 intersecting lines. Vertical angles are always congruent. In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles.
Course 3 Points, Lines, Planes, and Angles Example: Finding the Measure of Vertical Angles In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. If m1 = 37°, find m3. The measures of 1 and 2 are supplementary. The measures of 2 and 3 are supplementary. m2 = 180° – 37° = 143° m3 = 180° – 143° = 37° So m1 = m3 or 1 = 3. ~
Course 3 Parallel and Perpendicular Lines Parallel lines are lines in a plane that never meet, like a set of perfectly straight, infinite train tracks. The symbol for parallel is ||.
Course 3 Parallel and Perpendicular Lines The railroad ties are transversals to the tracks. A transversal is a line that intersects 2 or more lines in the same plane. It creates angles with special properties. The tracks are parallel.
Course 3 Parallel and Perpendicular Lines Example: Identifying Congruent Angles Formed by a Transversal Look at the angles formed by the transversal and parallel lines. Which angles seem to be congruent? 1, 3, 5, and 7 all measure 150°. 2, 4, 6, and 8 all measure 30°.
Course 3 Parallel and Perpendicular Lines Example Continued Angles marked in blue appear to be congruent to each other, and angles marked in red appear to be congruent to each other. 1 3 5 7 2 4 6
Course 3 Parallel and Perpendicular Lines Alternate interior angles: 2 angles on opposite sides of the transversal and inside the parallel lines. These angles are ≌. The pair of blue and the pair of pink angles are alternate interior angles.
Course 3 Parallel and Perpendicular Lines Alternate exterior angles: 2 angles on opposite sides of the transversal and outside the parallel lines. These angles are ≌. The pair of blue and the pair of pink angles are alternate exterior angles.
Course 3 Parallel and Perpendicular Lines Corresponding angles: Angles in matching positions when 2 parallel lines are crossed by a transversal. Corresponding angles are ≌. The pair of pink angles are corresponding. The pair of purple angles are corresponding. The light blue pairs and green pairs are also corresponding.
Course 3 Parallel and Perpendicular Lines Same side interior or consecutive interior angles are 2 angles inside the 2 parallel lines along the same side of a transversal line. These angles are supplementary Ex: <3 and <5 are same side interior angles. <4 and <6 are same side interior angles.
Course 3 Parallel and Perpendicular Lines Same side exterior or consecutive exterior angles are 2 angles outside the 2 parallel lines along the same side of a transversal line. These angles are supplementary. Ex: <1 and <7 are same side exterior angles. <2 and <8 are same side exterior angles
Course Parallel and Perpendicular Lines PROPERTIES OF TRANSVERSALS TO PARALLEL LINES If two parallel lines are intersected by a transversal, the acute angles that are formed are all congruent, the obtuse angles are all congruent, and any acute angle is supplementary to any obtuse angle. If the transversal is perpendicular to the parallel lines, all angles are 90°.
In the figure, line l (L) || line m. Find the measure of the angle. Course 3 Parallel and Perpendicular Lines Example: Finding Angle Measures of Parallel Lines Cut by Transversals 44 m 4 = 124° All obtuse angles in the figure are congruent.
Course 3 Parallel and Perpendicular Lines Example: Finding Angle Measures of Parallel Lines Cut by Transversals Continued 22 m ° = 180° 2 is supplementary to the angle 124°. m2 = 56° –124° In the figure, line l || line m. Find the measure of the angle.
Course 3 Parallel and Perpendicular Lines Example: Finding Angle Measures of Parallel Lines Cut by Transversals Continued Every acute angle is supplementary to every obtuse angle. 66 m 6 = 56° In the figure, line l || line m. Find the measure of the angle.
In the figure, line n || line m. Find the measure of the angle. Course 3 Parallel and Perpendicular Lines Example 77 m 7 = 144° All obtuse angles in the figure are congruent 1 144° m n
Course 3 Parallel and Perpendicular Lines 55 m ° = 180° 5 is supplementary to the angle 144°. m5 = 36° –144° 1 144° m n In the figure, line n || line m. Find the measure of the angle. Example
Course 3 Parallel and Perpendicular Lines Every acute angle is supplementary to every obtuse angle. 11 180°-144°=36° m 1 = 36° 1 144° m n In the figure, line n || line m. Find the measure of the angle. Example
Course 3 Parallel and Perpendicular Lines Checkpoint Quiz In the figure a || b. 1. Name the angles congruent to 3. 2. Name all the angles supplementary to 6. 3. If m1 = 105° what is m3? 4. If m5 = 120° what is m2? 1, 5, 7 1, 3, 5, 7 105° 60°
Course 3 Points, Lines, Planes, and Angles Checkpoint Quiz In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. 1. Name three points in the figure. 3. Name a right angle in the figure. 4. Name a pair of complementary angles. 5. If m1 = 47°, then find m3. 2. Name two lines in the figure. Possible answer: A, B, and C Possible answer: AGF Possible answer: 1 and 2 47° Possible answer: AD and BE