ANGULAR GEOMETRIC PRINCIPLES Unit 21 ANGULAR GEOMETRIC PRINCIPLES

NAMING ANGLES Angles are named by a number, a letter, or three letters For example, the angle shown can be called 1, B, ABC, or CBA When an angle is named with three letters, the vertex must be the middle letter. In cases where a point is the vertex of more than one angle, a single letter cannot be used to name an angle A B C 1

TYPES OF ANGLES An acute angle is an angle that is less than 90° A right angle is an angle of 90° An obtuse angle is an angle greater than 90° and less than 180° A straight angle is an angle of 180° A reflex angle is an angle greater than 180° and less than 360° Two angles are adjacent if they have a common vertex and a common side

ANGLES FORMED BY A TRANSVERSAL Transversal: A line that intersects (cuts) two or more lines. Alternate interior angles: Pairs of interior angles on opposite sides of the transversal and have different vertices. Corresponding angles: Pairs of angles, one interior and one exterior, located on same side of the transversal, but with different vertices.

ANGLES CREATED BY A TRANSVERSAL 1 2 3 4 5 6 7 8 l is the transversal angles 3 & 5 angles 4 & 6 are pairs of alternate interior angles

ANGLES CREATED BY A TRANSVERSAL l is the transversal 1 2 4 3 angles 1 & 5, 2 & 6, 3&7, 4&8 are pairs of corresponding angles 5 6 8 7

THEOREMS AND COROLLARIES A theorem is a statement in geometry that can be proved. A corollary is a statement based on a theorem. A corollary is often a special case of a theorem

THEOREMS AND COROLLARIES (Cont) The following are theorems and corollaries used throughout the text. They are numbered for easier reference (follow in your book for a visual) If two lines intersect, the opposite, or vertical angles are equal. If two parallel lines are intersected by a transversal, the alternate interior angles are equal If two lines are intersected by a transversal and a pair of alternate interior angles are equal, the lines are parallel.

THEOREMS AND COROLLARIES (Cont) If two parallel lines are intersected by a transversal, the corresponding angles are equal If two lines are intersected by a transversal and a pair of corresponding angles are equal, the lines are parallel Two angles are either equal or supplementary if their corresponding sides are parallel Two angles are either equal or supplementary if their corresponding sides are perpendicular

ANGULAR MEASURE EXAMPLE Determine the measure of all the missing angles in the figure below given that l  m, p  q, 1 = 110°, and 2 = 80°: q 3 = 110 because it is vertical to 1 (Theorem #1) 1 2 l 5 3 4 = 110 because it is alternate interior to 3 (Theorem #2) and corresponding to 1 (Theorem #4) 4 8 m 6 p 5 = 70 (180° – 110°) because it is supplementary to both 1 and 3

ANGULAR MEASURE EXAMPLE Determine the measure of all the missing angles in the figure below given that l  m, p  q, 1 = 110°, and 2 = 80°: q 6 = 70 because it is corresponding to 5 (Theorem #4) 1 2 l 5 3 8 = 2 = 80 because two angles are either equal or supplementary if their corresponding sides are parallel (Theorem #6) 4 8 m 6 p

PRACTICE PROBLEMS Define the terms in problems 1–6: Obtuse angle Reflex angle Corresponding angles Transversal Straight angle Name 1 in the figure below in three additional ways: C D E 1

PRACTICE PROBLEMS (Cont) Determine the measure of angles 2–8 in the figure below given that l  m and that 1 = 50° l m 1 2 3 4 5 6 7 8

PROBLEM ANSWER KEY An angle greater than 90 and less than 180 A pair of angles, one interior and one exterior. Both angles are on the same side of the transversal with different vertices A line that intersects (cuts) two or more lines An angle of 180° D, CDE, EDC 2 = 130, 3 = 50, 4 = 130, 5 = 130, 6 = 50, 7 = 50, and 8 = 130