* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Angles-and-Their-Measures
Line (geometry) wikipedia , lookup
Rotation formalisms in three dimensions wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Integer triangle wikipedia , lookup
Technical drawing wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Multilateration wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Trigonometric functions wikipedia , lookup
Angles and Their Measures Standard/Objectives: Standard: Students will understand geometric concepts and applications Benchmark: Use visualization, spatial reasoning, and geometric modeling to solve problems. Standard/Objectives: Performance Standard: Solve problems involving complementary, supplementary and congruent angles. Objectives: • Use angle postulates • Classify angles as acute, right, obtuse, or straight. 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. vertex The point A is the vertex of the angle. C sides B A Ex.1: Naming Angles • Name the angles in the figure: S SOLUTION: There are three Q different angles. R • PQS or SQP You should not name any of • SQR or RQS these angles as Q because • PQR or RQP all three angles have Q as their P vertex. The name Q would not distinguish one angle from the others. Note: • The measure of A is denoted by mA. 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 mBAC = 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 50° congruent. E D F Note – Geometry doesn’t use equal signs like Algebra MEASURES ARE EQUAL ANGLES ARE CONGRUENT mBAC = mDEF 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 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 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 Postulate 4: Angle Addition Postulate • If P is in the interior of RST, then mRSP + mPST = mRST R P S T 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. Solution: You can use the Angle Addition Postulate. 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. 3: 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. 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.