Transcription
7.1Adjacent and Vertical AnglesWhat can you conclude about the anglesformed by two intersecting lines?Classification of AnglesAcute:Right:Obtuse:Straight:Less than 90 Equal to 90 Greater than 90 andless than 180 Equal to 180 1ACTIVITY: Drawing AnglesWork with a partner.a. Draw the hands of the clock to represent the given type of angle.AcuteStraightRightObtuseb. What is the measure of the angle formed by the hands of the clock at thegiven time?9:006:0012:00AdjacentGeometryIn this lesson, you will identify adjacent andvertical angles. find angle measuresusing adjacent andvertical angles.When two states are adjacent,MaineNew Hampshire270Chapter 7ms red pe 0701.indd 270they are next to each other and theyshare a common border.MaineNew HampshireConstructions and Scale Drawings2/2/15 8:21:26 AM
2ACTIVITY: Naming AnglesWork with a partner. Some angles, such as A, can be named by a singleletter. When this does not clearly identify an angle, you should use threeletters, as shown.MathPractice BDEpoint on one sideJustifyConclusionspoint on other sidevertexWhen you name anangle, does the orderin which you writethe letters matter?Explain.BACƋAABEF andBCDE aresquares.ƋBDEFEDa. Name all the right angles, acute angles, and obtuse angles.b. Which pairs of angles do you think are adjacent? Explain.3ACTIVITY: Measuring AnglesontRkm832Oaa. How many angles are formed bythe intersecting roads? Numberthe angles.dWork with a partner.b. CHOOSE TOOLS Measure each angleformed by the intersecting roads.What do you notice?8324. IN YOUR OWN WORDS What can you conclude about the anglesformed by two intersecting lines?5. Draw two acute angles that are adjacent.Use what you learned about angles and intersecting lines tocomplete Exercises 3 and 4 on page 274.Section 7.1ms red pe 0701.indd 271Adjacent and Vertical Angles2712/2/15 8:21:39 AM
Lesson7.1Lesson TutorialsKey Vocabularyadjacent angles,p. 272vertical angles, p. 272congruent angles,p. 272Adjacent AnglesWordsTwo angles are adjacent angles when they share a commonside and have the same vertex. 1 and 2 are adjacent.Examples 2 and 4 are not adjacent.1 24 3Vertical AnglesWordsTwo angles are vertical angles when they are opposite anglesformed by the intersection of two lines. Vertical anglesare congruent angles, meaning they have the same measure.Examples 1 and 3 are vertical angles.14 2 and 4 are vertical angles.23EXAMPLE1Naming AnglesUse the figure shown.a. Name a pair of adjacent angles. ABC and ABF share a common sideand have the same vertex B.FABSo, ABC and ABF are adjacent angles.Eb. Name a pair of vertical angles.DC ABF and CBD are opposite anglesformed by the intersection of two lines.So, ABF and CBD are vertical angles.Exercises 5 and 6Name two pairs of adjacent angles and two pairs of vertical anglesin the figure.1.2.VXQPWJZNY272Chapter 7ms red pe 0701.indd 272KLMConstructions and Scale Drawings2/2/15 8:21:54 AM
EXAMPLEUsing Adjacent and Vertical Angles2Tell whether the angles are adjacent or vertical. Then find thevalue of x.The angles are vertical angles.Because vertical angles are congruent,the angles have the same measure.a.70íxíSo, the value of x is 70.RememberThe angles are adjacent angles.Because the angles make up a rightangle, the sum of their measures is 90 .b.You can add anglemeasures. When twoor more adjacentangles form a largerangle, the sum ofthe measures of thesmaller angles is equalto the measure of thelarger angle.(x 4) 31 9031íx 35 90(x à 4)íx 55Write equation.Combine like terms.Subtract 35 from each side.So, the value of x is 55.EXAMPLE3Constructing AnglesDraw a pair of vertical angles with a measure of 40 .Exercises 8 –17Step 2: Use a straightedge to extendthe sides to form twointersecting lines.80 90 10 070 10 0 90 80 110 170 2060 0 11060 12350 0 150 01340í170 180600 1 20 10 0150 3014 040 10180 170 1 20 36015 0 401 040Step 1: Use a protractor todraw a 40 angle.40í40íTell whether the angles are adjacent or vertical. Then find thevalue of x.3.4.85í xí5.xí69í(2x Ź 1)í6. Draw a pair of vertical angles with a measure of 75 .Section 7.1ms red pe 0701.indd 273Adjacent and Vertical Angles2732/2/15 8:22:01 AM
Exercises7.1Help with Homework1. VOCABULARY When two lines intersect, how many pairsof vertical angles are formed? How many pairs of adjacentangles are formed?1 24 32. REASONING Identify the congruent angles in the figure.Explain your reasoning.6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Use the figure at the right.CA3. Measure each angle formed by the intersecting lines.BD4. Name two angles that are adjacent to ABC.EName two pairs of adjacent angles and two pairs of vertical angles in the figure.16.5.HFLNPMGSKJQR7. ERROR ANALYSIS Describe andcorrect the error in naming a pairof vertical angles. BDCAEF ACB and BCDare vertical angles.Tell whether the angles are adjacent or vertical. Then find the value of (4x Ź 25)í(x à 20)í4xí2xí274Chapter 7ms red pe 0701.indd 274Constructions and Scale Drawings2/2/15 8:22:04 AM
Draw a pair of vertical angles with the given measure.3 14. 25 15. 85 16. 110 17. 135 18. IRON CROSS The iron cross is a skiing trick in whichthe tips of the skis are crossed while the skier isairborne. Find the value of x in the iron cross shown.19. OPEN-ENDED Draw a pair of adjacent angles with thegiven description.a. Both angles are acute.127í(2x à 41)íb. One angle is acute, and one is obtuse.c. The sum of the angle measures is 135 .20. PRECISION Explain two procedures that you can use to draw adjacent angleswith given measures.Determine whether the statement is always, sometimes, or never true.21. When the measure of 1 is 70 ,the measure of 3 is 110 .1 24 322. When the measure of 4 is 120 ,the measure of 1 is 60 .23. 2 and 3 are congruent.24. The measure of 1 plus the measure of 2 equalsthe measure of 3 plus the measure of 4.25. REASONING Draw a figure in which 1 and 2 are acutevertical angles, 3 is a right angle adjacent to 2, and thesum of the measure of 1 and the measure of 4 is 180 .26.For safety reasons, a ladder should makea 15 angle with a wall. Is the ladder shown leaning ata safe angle? Explain.120íSolve the inequality. Graph the solution. (Section 4.3)1227. 6n 54m 2.528. —x 1729. 1.6 —30. MULTIPLE CHOICE What is the slope of the line that passes through the points(2, 3) and (6, 8)? (Section 5.5)A 45—B 54—C Section 7.1ms red pe 0701.indd 27543—D 32—Adjacent and Vertical Angles2752/2/15 8:22:08 AM
7.1 Lesson Lesson Tutorials Key Vocabulary adjacent angles, p. 272 vertical angles, p. 272 congruent angles, p. 272 EXAMPLE 1 Naming Angles Use the fi gure shown. a. Name a pair of adjacent angles. A B D E C ABC and ABF share a common side F and have the same vertex B. So ,
