Obj. 8 Classifying Angles and Pairs of Angles
- 1. Obj. 8 Angles Objectives: The student will be able to (I can): Correctly name an angle Classify angles as acute, right, or obtuse IdentifyIdentify • linear pairs • vertical angles • complementary angles • supplementary angles and set up and solve equations. Obj. 8 Angles The student will be able to (I can): Classify angles as acute, right, or obtuse and set up and solve equations.
- 2. angle vertex A figure formed by two rays or sides with a common endpoint. Example: The common endpoint of two rays or sides (plural vertices(plural vertices Example: A is the vertex of the above angle A figure formed by two rays or sides with a common endpoint. The common endpoint of two rays or sides vertices). ● ● ● A C R vertices). Example: A is the vertex of the above angle
- 3. Notation: An angle is named one of three different ways: 1. By the vertex and a point on each ray1. By the vertex and a point on each ray (vertex must be in the middle) : 2. By its vertex (if only one angle): 3. By a number: Notation: An angle is named one of three different ways: 1. By the vertex and a point on each ray ● ● ● E T A 1 1. By the vertex and a point on each ray (vertex must be in the middle) : ∠TEA or ∠AET By its vertex (if only one angle): ∠E 3. By a number: ∠1
- 4. Example Which name is below? ∠TRS∠TRS ∠SRT ∠RST ∠2 ∠R Which name is notnotnotnot correct for the angle ● ● ● S R T 2
- 5. Example Which name is below? ∠TRS∠TRS ∠SRT ∠RST ∠2 ∠R Which name is notnotnotnot correct for the angle ● ● ● S R T 2
- 6. acute angle right angle Angle whose measure is greater than 0º and less than 90º. Angle whose measure is exactly 90º. obtuse angle Angle whose measure is greater than 90º and less than 180º. Angle whose measure is greater than 0º and less than 90º. Angle whose measure is exactly 90º. Angle whose measure is greater than 90º and less than 180º.
- 7. congruent angles Angles that have the same measure. m∠WIN = m∠ ∠WIN ≅ ∠LHS N Notation: “Arc marks” indicate congruent angles. Notation: To write the measure of an angle, put a lowercase “m” in front of the angle bracket. m∠WIN is read “measure of angle WIN” Angles that have the same measure. ∠LHS LHS ●● ● ● ● ● L H S W IN Notation: “Arc marks” indicate congruent Notation: To write the measure of an angle, put a lowercase “m” in front of the angle WIN is read “measure of angle WIN”
- 8. interior of an angle Angle Addition Postulate The set of all points between the sides of an angle If D is in the interiorinteriorinteriorinterior m∠ABD + m (part + part = whole) Example: If m m∠ ● A The set of all points between the sides of interiorinteriorinteriorinterior of ∠ABC, then ABD + m∠DBC = m∠ABC (part + part = whole) ● Example: If m∠ABD=50˚ and ∠ABC=110˚, then m∠DBC=60˚ ● ● ● B D C
- 9. Example The m∠PAH = 125˚. ● P (2x+8) PAH = 125˚. Solve for x. ● ● ● A T H (3x+7)˚ (2x+8)˚
- 10. Example The m∠PAH = 125˚. m∠PAT + m∠ ● P (2x+8) m∠PAT + m∠ PAH = 125˚. Solve for x. ∠TAH = m∠PAH ● ● ● A T H (3x+7)˚ (2x+8)˚ ∠TAH = m∠PAH
- 11. Example The m∠PAH = 125˚. m∠PAT + m∠ ● P (2x+8) m∠PAT + m∠ 2x + 8 + 3x + 7 = 125 PAH = 125˚. Solve for x. ∠TAH = m∠PAH ● ● ● A T H (3x+7)˚ (2x+8)˚ ∠TAH = m∠PAH 2x + 8 + 3x + 7 = 125
- 12. Example The m∠PAH = 125˚. m∠PAT + m∠ ● P (2x+8) m∠PAT + m∠ 2x + 8 + 3x + 7 = 125 5x + 15 = 125 PAH = 125˚. Solve for x. ∠TAH = m∠PAH ● ● ● A T H (3x+7)˚ (2x+8)˚ ∠TAH = m∠PAH 2x + 8 + 3x + 7 = 125 5x + 15 = 125
- 13. Example The m∠PAH = 125˚. m∠PAT + m∠ ● P (2x+8) m∠PAT + m∠ 2x + 8 + 3x + 7 = 125 5x + 15 = 125 PAH = 125˚. Solve for x. ∠TAH = m∠PAH ● ● ● A T H (3x+7)˚ (2x+8)˚ ∠TAH = m∠PAH 2x + 8 + 3x + 7 = 125 5x + 15 = 125 5x = 110
- 14. Example The m∠PAH = 125˚. m∠PAT + m∠ ● P (2x+8) m∠PAT + m∠ 2x + 8 + 3x + 7 = 125 5x + 15 = 125 PAH = 125˚. Solve for x. ∠TAH = m∠PAH ● ● ● A T H (3x+7)˚ (2x+8)˚ ∠TAH = m∠PAH 2x + 8 + 3x + 7 = 125 5x + 15 = 125 5x = 110 x = 22
- 15. angle bisector A ray that divides an angle into two congruent angles. Example: UY bisects ∠SUN; thusUY bisects ∠SUN; thus A ray that divides an angle into two congruent angles. SUN; thus ∠SUY ≅ ∠YUN ● ●● ● S U N Y SUN; thus ∠SUY ≅ ∠YUN or m∠SUY = m∠YUN
- 16. adjacent angles Two angles in the same plane with a common vertex and a common side, but no common interior points. Example: ∠1 and ∠2 are adjacent angles. linear pair ∠1 and ∠2 are adjacent angles. Two adjacent angles whose noncommon sides are opposite rays. (They form a line.) Example: Two angles in the same plane with a common vertex and a common side, but no common interior points. 2 are adjacent angles. 1 2 2 are adjacent angles. Two adjacent angles whose noncommon sides are opposite rays. (They form a line.)
- 17. vertical angles Two nonadjacent angles formed by two intersecting lines. congruent.congruent.congruent.congruent. Example: ∠1 and ∠2 and Two nonadjacent angles formed by two intersecting lines. They are alwaysThey are alwaysThey are alwaysThey are always 1 2 3 4 1 and ∠4 are vertical angles 2 and ∠3 are vertical angles
- 18. complementary angles supplementary angles Two angles whose measures have the sum of 90º. Two angles whose measures have the sum of 180º. ∠A and ∠B are complementary. (55+35) ∠A and ∠C are supplementary. (55+125) Two angles whose measures have the sum Two angles whose measures have the sum 55º 35º B are complementary. (55+35) C are supplementary. (55+125) A B C 125º
- 19. Practice 1. What is m 2. What is m 3. What is m What is m∠1? What is m∠2? 1 60˚ What is m∠3? 51˚ 2 105˚ 3
- 20. Practice 1. What is m 180 — 60 = 120 2. What is m 3. What is m What is m∠1? 60 = 120˚ What is m∠2? 1 60˚ What is m∠3? 51˚ 2 105˚ 3
- 21. Practice 1. What is m 180 — 60 = 120 2. What is m 90 — 51 = 39 3. What is m What is m∠1? 60 = 120˚ What is m∠2? 51 = 39˚ 1 60˚ What is m∠3? 51˚ 2 105˚ 3
- 22. Practice 1. What is m 180 — 60 = 120 2. What is m 90 — 51 = 39 3. What is m 105˚ What is m∠1? 60 = 120˚ What is m∠2? 51 = 39˚ 1 60˚ What is m∠3? 51˚ 2 105˚ 3