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# Angles and their measures

mathematics

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### Angles and their measures

1. 1. Angles and Their Measures
2. 2. Using Angle Postulates • An angle consists of two different rays that have the same initial point. The rays are the sides of the angle. The initial point is the vertex of the angle. • The angle that has sides AB and AC is denoted by ∠BAC, ∠CAB, ∠A. The point A is the vertex of the angle. sides vertex C A B
3. 3. Ex.1: Naming Angles • Name the angles in the figure: SOLUTION: There are three different angles. ∀ ∠PQS or ∠SQP ∀ ∠SQR or ∠RQS ∀ ∠PQR or ∠RQP Q P S R You should not name any of these angles as ∠Q because all three angles have Q as their vertex. The name ∠Q would not distinguish one angle from the others.
4. 4. Note: • The measure of ∠A is denoted by m∠A. The measure of an angle can be approximated using a protractor, using units called degrees(°). For instance, ∠BAC has a measure of 50°, which can be written as m∠BAC = 50°. B A C
5. 5. more . . . • Angles that have the same measure are called congruent angles. For instance, ∠BAC and ∠DEF each have a measure of 50°, so they are congruent. D E F 50°
6. 6. Note – Geometry doesn’t use equal signs like Algebra MEASURES ARE EQUAL m∠BAC = m∠DEF ANGLES ARE CONGRUENT ∠BAC ≅ ∠DEF “is equal to” “is congruent to” Note that there is an m in front when you say equal to; whereas the congruency symbol ≅ ; you would say congruent to. (no m’s in front of the angle symbols).
7. 7. Postulate 3: Protractor Postulate • Consider a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers from 1- 180. • The measure of ∠AOB is equal to the absolute value of the difference between the real numbers for OA and OB. A O B
8. 8. A D E Interior/Exterior • A point is in the interior of an angle if it is between points that lie on each side of the angle. • A point is in the exterior of an angle if it is not on the angle or in its interior.
9. 9. Postulate 4: Angle Addition Postulate • If P is in the interior of ∠RST, then m∠RSP + m∠PST = m∠RST R S T P
10. 10. Ex. 2: Calculating Angle Measures • VISION. Each eye of a horse wearing blinkers has an angle of vision that measures 100°. The angle of vision that is seen by both eyes measures 60°. • Find the angle of vision seen by the left eye alone.
11. 11. Solution: You can use the Angle Addition Postulate.
12. 12. Classifying Angles • Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less than or equal to 180°.
13. 13. Ex. 3: Classifying Angles in a Coordinate Plane • Plot the points L(-4,2), M(-1,-1), N(2,2), Q(4,-1), and P(2,-4). Then measure and classify the following angles as acute, right, obtuse, or straight. α. ∠LMN β. ∠LMP χ. ∠NMQ δ. ∠LMQ
14. 14. Solution: • Begin by plotting the points. Then use a protractor to measure each angle.
15. 15. Solution: • Begin by plotting the points. Then use a protractor to measure each angle. Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.
16. 16. Ex. 4: Drawing Adjacent Angles • Use a protractor to draw two adjacent acute angles ∠RSP and ∠PST so that ∠RST is (a) acute and (b) obtuse.
17. 17. Ex. 4: Drawing Adjacent Angles • Use a protractor to draw two adjacent acute angles ∠RSP and ∠PST so that ∠RST is (a) acute and (b) obtuse.
18. 18. Ex. 4: Drawing Adjacent Angles • Use a protractor to draw two adjacent acute angles ∠RSP and ∠PST so that ∠RST is (a) acute and (b) obtuse. Solution:
19. 19. Closure Question: • Describe how angles are classified. Angles are classified according to their measure. Those measuring less than 90° are acute. Those measuring 90° are right. Those measuring between 90° and 180° are obtuse, and those measuring exactly 180° are straight angles.
• #### jen_2268

Nov. 29, 2016

mathematics

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