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Unit 21

Unit 21. ANGULAR GEOMETRIC PRINCIPLES. A. 1. B. C. 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.

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Unit 21

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  1. Unit 21 ANGULAR GEOMETRIC PRINCIPLES

  2. A 1 B C 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

  3. 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

  4. 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.

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

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

  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

  8. 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.

  9. 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

  10. 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

  11. 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

  12. C D 1 E 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:

  13. 2 3 l 1 4 5 6 7 8 m PRACTICE PROBLEMS (Cont) • Determine the measure of angles 2–8 in the figure below given that l m and that 1 = 50°

  14. PROBLEM ANSWER KEY • An angle greater than 90 and less than 180 • An angle greater than 180° and less than 360° • 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

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