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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.
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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.
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Name the three angles in the diagram.
EXAMPLE 1 Name angles Name the three angles in the diagram. WXY, or YXW YXZ, or ZXY WXZ, or ZXW You should not name any of these angles X because all three angles have X as their vertex.
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Extra 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.
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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
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Measure and classify angles
EXAMPLE 2 Measure and classify angles Use the diagram to find the measure of the indicated angle. Then classify the angle. a KHJ b GHK c GHJ d GHL SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.
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Measure and classify angles
EXAMPLE 2 Measure and classify angles a. HJ is lined up with the 0 on the inner scale of the protractor. HK passes through 55 on the inner scale. So, m KHJ = 55 . It is an acute angle. o b. HG is lined up with the 0 on the outer scale and HK passes through 125 on the outer scale. So, m GHK = It is an obtuse angle. o c. m GHJ = 180. It is a straight angle. o d. m GHL= 90. It is a right angle. o
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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
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TAKE NOTE! MEASURES ARE EQUAL mBAC = mDEF ANGLES ARE CONGRUENT
BAC DEF “is congruent to” “is equal 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).
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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
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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.
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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
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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°
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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.
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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
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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
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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. LMN LMP NMQ LMQ
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Solution: Begin by plotting the points. Then use a protractor to measure each angle.
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Solution: Begin by plotting the points. Then use a protractor to measure each angle.
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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!
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1. Name all the angles in the diagram. Which angle is a right angle?
GUIDED PRACTICE for Examples 1and 2 Name all the angles in the diagram. Which angle is a right angle? PQR , PQS, RQS ; PQS is a right angle . ANSWER
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2. Draw a pair of opposite rays. What type of angle do the rays form?
GUIDED PRACTICE for Examples 1and 2 Draw a pair of opposite rays. What type of angle do the rays form? ANSWER Straight Angle
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Angle Addition Postulate Substitute angle measures.
EXAMPLE 3 Find angle measures o ALGEBRA Given that m LKN =145 , find m LKM and m MKN. SOLUTION STEP 1 Write and solve an equation to find the value of x. m LKN = m LKM + m MKN Angle Addition Postulate 145 = (2x + 10) + (4x – 3) o Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6.
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Evaluate the given expressions when x = 23.
EXAMPLE 3 Find angle measures STEP 2 Evaluate the given expressions when x = 23. m LKM = (2x + 10)° = ( )° = 56° m MKN = (4x – 3)° = (4 23 – 3)° = 89° So, m LKM = 56° and m MKN = 89°. ANSWER
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Find the indicated angle measures.
GUIDED PRACTICE for Example 3 Find the indicated angle measures. Given that KLM is a straight angle, find m KLN and m NLM. ANSWER 125°, 55°
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4. Given that EFG is a right angle, find m EFH and m HFG.
GUIDED PRACTICE for Example 3 4. Given that EFG is a right angle, find m EFH and m HFG. ANSWER 60°, 30°
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Identify congruent angles
EXAMPLE 4 Identify congruent angles The diagram shows some of the angles formed by the ropes in a trapeze apparatus. Identify the congruent angles. If m DEG = 157° ,what is m GKL? Trapeze SOLUTION There are two pairs of congruent angles: DEF JKL and DEG GKL. ~ Because DEG GKL, m DEG = m GKL. So, m GKL = 157°. ~
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5. Identify all pairs of congruent angles in the diagram.
GUIDED PRACTICE for Example 4 Use the diagram shown. 5. Identify all pairs of congruent angles in the diagram. T and S, P and R. ANSWER
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GUIDED PRACTICE for Example 4 Use the diagram shown.
6. In the diagram, m PQR = 130 , m QRS = 84, and m TSR = Find the other angle measures in the diagram. o m PTS = 121, m QPT = 84° ANSWER
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An ANGLE BISECTOR is a ray that divides an angle into two angles that are congruent.
Draw and label an acute angle, Fold the paper so that MN is on top of LM. Draw a point P on the folded side. Connect it with M. Measure the angles formed.
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Double an angle measure
EXAMPLE 5 Double an angle measure In the diagram at the right, YW bisects XYZ, and m XYW = 18 . Find m XYZ. o SOLUTION By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~ So, m XYW = m WYZ, and you can write m XYZ = m XYW + m WYZ = 18°+ 18° = 36°.
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GUIDED PRACTICE for Example 5 7. Angle MNP is a straight angle, and NQ bisects MNP. Draw MNP And NQ . Use arcs to mark the congruent angles in your diagram, and give the angle measures of these congruent angles. 90° ANSWER
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Classify each angle as acute, obtuse , right or straight.
Daily Homework Quiz Classify each angle as acute, obtuse , right or straight. o 1. m A = 90 ANSWER right 2. m B = 62 o ANSWER acute 3. m C = 119 o ANSWER obtuse
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5. If XY bisects ZXW and m XZY = 36 , find m ZXW.
Daily Homework Quiz If m DEG = 84 , find m FEG. o ANSWER 14 o 5. If XY bisects ZXW and m XZY = 36 , find m ZXW. o ANSWER 72 o
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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
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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. LMN LMP NMQ LMQ
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Solution: Begin by plotting the points. Then use a protractor to measure each angle.
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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.
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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
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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?
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QUESTIONS?
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