Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rotation formalisms in three dimensions wikipedia , lookup
Integer triangle wikipedia , lookup
Pythagorean theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Perceived visual angle wikipedia , lookup
1.4: Measure and Classify Angles 1.5: Describe Angle Pair Relationships Objectives: 1. To define, classify, draw, name, and measure various angles 2. To use the Protractor and Angle Addition Postulates 3. To use special angle relationships to find angle measures Magic Dough Revisited Keeping in mind the rules of working with Magic Dough, create an angle with your dough. Don’t forget to put the arrows and points in the appropriate places. Angle Vertex A Sides B C • An angle consists of two different rays (sides) that share a common endpoint (vertex). – Angle ABC, ABC, or B A “Rabbit Ear” antenna is a physical model of an angle Angle • An angle consists of two different rays (sides) that share a common endpoint (vertex). – Angle ABC, ABC, or B Example 1 How many angles can be seen in the diagram? Name all the angles. W Y X Z How Big is an Angle? Is the angle between the two hands of the wristwatch smaller than the angle between the hands of the large clock? – Both clocks read 9:36 Click me to learn more about measuring angles Measure of an Angle The measure of an angle is the smallest amount of rotation about the vertex from one side to the other, measured in degrees. • Can be any value between 0 and 180 • Measured with a protractor Classifying Angles Create an example of each of these angles with your Magic Dough. How To Use a Protractor The measure of this angle is written: mABC 34 Example 2 Use the diagram to fine the measure of the indicated angle. Then classify the angle. 1. KHJ 2. GHK 3. GHJ 4. GHL Example 3 Turn to page 28 of your textbook. Use your protractor to measure the angles shown for exercises 3-5. Example 4 What is the measure of DOZ? D G 25 O 40 Z Example 4 You basically used the Angle Addition Postulate to get the measure of the angle, where mDOG + mGOZ = mDOZ. D G 25 O 40 Z Angle Addition Postulate If P is in the interior of RST, then mRST = mRSP + mPST. Example 5 Given that mLKN = 145°, find mLKM and mMKN. M L 2x+10 4x-3 K N Congruent Angles • Two angles are congruent angles if they have the same measure. Add the appropriate markings to your picture. Patty Paper Activity On a piece of patty paper, use a straightedge to draw and label acute angle ABC. Patty Paper Activity Fold the paper so that BC coincides with BA. Patty Paper Activity Unfold the paper and draw point D on the crease in the interior of ABC. What do you notice about ABD and DBC. BD is called an angle bisector. Angle Bisector An angle bisector is a ray that divides an angle into two congruent angles. Example 6 In the diagram, YW bisects XYZ, and mXYW = 18°. Find mXYZ. X W Y Z Angle Pair Investigation In this Investigation, you will be shown examples and non-examples of various angle pairs. Use the pictures to come up with a definition of each angle pair. Complementary Angles Supplementary Angles C Comes Before S… m1 m2 90 m3 m4 90 m5 m6 180 m7 m8 180 Linear Pairs of Angles Linear Pairs of Angles • Two adjacent angles form a linear pair if their noncommon sides are opposite rays. • The angles in a linear pair are supplementary. Vertical Angles Vertical Angles • Two nonadjacent angles are vertical angles if their sides form two pairs of opposite rays. • Vertical angles are formed by two intersecting lines. Example 7 1. Given that 1 is a complement of 2 and m1 = 68°, find m2. 2. Given that 3 is a complement of 4 and m3 = 56°, find m4. Example 8 Identify all of the linear pairs of angles and all of the vertical angles in the figure. Example 9 Two angles form a linear pair. The measure of one angle is 5 times the measure of the other angle. Find the measure of each angle.