1.4 What you should learn Why you should learn it Angles and Their Measures What you should learn GOAL 1 Use Angle Postulates GOAL 2 Classify angles as acute, right, obtuse, or straight. Why you should learn it To solve real-life problems about angles, such as the field of vision of a horse wearing blinkers.

1.4 Angles and Their Measures 1 GOAL USING ANGLE POSTULATES An _____ consists of two different rays that have the same initial point. angle The rays are the _____ of the angle and the initial point is the ______. sides vertex The three names for this angle are A C B The sides are The vertex is point A.

Extra Example 1 Did you study Example 1??? Name the angles in the figure. M N O P Click for the answers. should not be used to name any angle in the figure. Why not? 1. 2. 3. All three angles have N as the vertex, so could mean any of the angles.

Measuring Angles The expression is read as “the ________ of angle A.” measure IMPORTANT!!! Note the difference in notation between an angle and its measure. Always use the correct notation!!! The tool used to measure angles is called a _________. protractor The units used to measure angles are called _______, and the symbol for them is a _. degrees

Measuring Angles Q R S Let’s measure some angles. Y M T Since we say that the angles are _________. congruent Remember: Angles are congruent, measures are equal. Incorrect: Correct: and

PROTRACTOR POSTULATE For any point A on one side of , can be matched one to one with the real numbers from 0 to ___. 180 The absolute value of the difference between the real numbers for is the ________ of measure Check your understanding of the Protractor Postulate by finding Use either scale on the protractor to find it, but use the same one for both rays. B O A Click to see two solutions. or

To understand the next postulate, you must understand some vocabulary: A point that is between points that lie on each side of an angle is in the _______ of the angle. A point that is not on an angle or in its interior is in the ________ of the angle. interior exterior exterior B interior A Z In the above diagram, A is in the interior of the angle and B is in the exterior of the angle.

ANGLE ADDITION POSTULATE If P is in the interior of then R T P S Does it make sense? Study the figure to be sure you understand before going on! EXAMPLE 2

What is the measure of the angle that is shaded green? Extra Example 2 The backyard of a house is illuminated by a light fixture that has two bulbs. Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs? Click for a picture. What is the measure of the angle that is shaded green? 120° Click for the solution. 120° 120° - 35° = 85° 85° Do you see the application of the Angle Addition Postulate? 35°

Checkpoint D Name the angles in the figure. Click for the solution. C In the figure above, and Find the measure of Click for the solution.

1.4 Angles and Their Measures 2 GOAL CLASSIFYING ANGLES We can classify all angles by their measure as follows: Angles with measures between 0 and 90 degrees: _____ 90 degree angles: _____ Angles with measures between 90 and 180 degrees: ______ 180 degree angles: _______ acute right obtuse straight

Can you match each angle with its description? Click to check your answer. A Important!!! Anytime this box appears at the vertex of an angle, it means that the angle is a right angle. You MUST use it when you want to show a right angle. acute: right: obtuse: straight: A A A EXAMPLE 3

Extra Example 3 Plot the points A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2). Then measure and classify the following angles as acute, right, obtuse, or straight. Click for each answer. a. b. c. d. 45°, acute 90°, right 180°, straight 135°, obtuse Reminder: Be sure you understand the material before going on to the next slide. If you need to review, do so NOW!

The common vertex is __ and the common side is ___. Two angles that share a common vertex and side, but have no common interior points, are called ________ angles. adjacent A B C D Name the two adjacent angles in the diagram. Click to check. The common vertex is __ and the common side is ___. D EXAMPLE 4

Extra Example 4 Use a protractor to draw two adjacent angles and so that is acute and is straight. Click to see a sample answer. L N M O Classify as acute, right, obtuse, or straight: obtuse

Does your solution meet the requirements above? Checkpoint Draw 5 points A, B, C, D, and E so that all four statements are true: are adjacent. is obtuse. D is in the exterior of is a right angle. Click to see a sample answer. B C E A D Does your solution meet the requirements above?

QUESTIONS?