1 4 Measuring Angles Using Angle Postulates Angle

1 -4 Measuring Angles

Using Angle Postulates • Angle- formed by two rays with the same endpoint. • The rays are the sides of the angle. The end 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.

Ex. 1: Naming Angles • Name the angles in the figure: SOLUTION: There are three different angles. • PQS or SQP You should not name any of • SQR or RQS these angles as Q because all • PQR or RQP three angles have Q as their vertex. The name Q would not distinguish one angle from the others.

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 B m BAC = 50°. A C

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. 50°

Note – Geometry doesn’t use equal signs like Algebra MEASURES ARE EQUAL ANGLES ARE CONGRUENT m BAC = m DEF 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).

Postulate 1 -7: 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 1180. • The measure of AOB is equal to the absolute value of the difference between the real numbers for OA and OB. A O B

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.

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°.

Ex. 2: Classifying Angles in a Coordinate Plane • a. b. c. d. 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. LMN LMP NMQ LMQ

Solution: • Begin by plotting the points. Then use a protractor to measure each angle.

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.

Postulate 1 -8: Angle Addition Postulate • If P is in the interior of RST, then m RSP + m PST = m RST

Ex. 3: 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.

Solution: You can use the Angle Addition Postulate.

Ex. 4 • m RST= 155, what is the measure of m RSP and m PST. ) 20 x 4 ( ) (3 x+14

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.

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.

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:

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.