BLoCK 1 ~ LInes And AngLes

Transcription

1 BLoCK ~ LInes And AngLes angle pairs Lesson MeasUring and naming angles Lesson classifying angles Explore! Classify an Angle Lesson 3 complementary and supplementary angles Explore! Complementary vs Supplementary Lesson 4 vertical angles Explore! The Vertical Angle Relationship Lesson 5 alternate exterior and interior angles Explore! Alternate Exterior and Alternate Interior Lesson 6 corresponding and same-side interior angles Explore! More Angle Pairs review BLock ~ angle Pairs obtuse Angle AlternAte exterior Angles word wall Acute Angle AdJAcent Angles same-side interior Angles AlternAte interior Angles ray supplementary Angles VerticAl Angles Angle transversal linear PAir degree complementary Angles right Angle congruent Protractor Vertex straight Angle corresponding Angles Block ~ Lines And Angles ~ Angle Pairs

2 BLoCK ~ AngLe PAIrs tic - tac - toe PuZZling Angles Bisecting Angles FliP BooK Find angle measures in a complex diagram. Use two types of constructions to bisect angles. Create a flip book which describes special angle pairs. Special Angle Pairs See page 34 for details. See page 8 for details. See page 33 for details. crossword Make a crossword using vocabulary from this block. ProtrActor guide Write a guide for using a protractor. transversal collage Find or take pictures of transversals and display them. See page 34 for details. See page for details. See page 7 for details. Angle Art BooK of Poetry duplicating Angles Create an original piece of artwork with lines and angles. Write poems about special angle pairs. Make an illustrated poetry booklet. Use a compass and straightedge to duplicate angles. See page 7 for details. See page for details. See page 7 for details. Block ~ Angle Pairs ~ Tic - Tac - Toe

3 measuring and naming angles Lesson Angles are used in construction, architecture, graphic design, aerospace, art, machining and manufacturing, as well as many other fields. An angle is formed by two rays with a common endpoint. A ray has one endpoint and extends forever in one direction. N NA NG (ray NG) A (ray NA) Ray NA is written NA. Ray NG is written NG. The first point in the name of a ray is the endpoint. Th e vertex of an angle is the common point of both rays. N is the vertex of this angle. When three points are used to name an angle, the vertex is written in the middle of the name. The vertex can be written as the name of an angle when it is the vertex for only one angle. G Three ways to name the angle formed by NA and NG are ANG, GNA and N. example give 4 different names for the given angle. P A L solutions. PAL. LAP 3. A 4. example Is W another name for nwe? explain. E S N W solution No, it is not clear whether W refers to NWE, SWE or NWS. Adjacent angles are two angles that share a ray. In Example, NWE and SWE share WE. This means NWE and SWE are adjacent angles. Lesson ~ Measuring And Naming Angles 3

4 example 3 Identify at least one additional name for each angle. Write using proper angle notation. a. T I b. TGI c. RGE E G 3 R solutions a. TGE or EGT b. or IGT c. EGR or 3 A protractor is a tool used to measure angles. Angles are measured in units called degrees. Th e m in front of an angle measure is notation for the word measure. The statement in Figure below reads, The measure of ABC is equal to sixty degrees. Figure Figure Figure 3 A Z P B C Q R X Y m ABC = 60 m PQR = 4 m ZYX = 74 4 Lesson ~ Measuring And Naming Angles

5 example 4 use a protractor to measure each angle. a. C b. A B 5 solutions C a. b. A B m ABC = 53 m 5 = 3 5 exercises give two different names for each angle Q L B P U sketch a diagram to represent each angle. J K A T 4. DOG 5. CUB also called 4 6. PAL also called 7. and are adjacent angles 8. XYZ and XYU are adjacent angles 9. HOT is approximately 90 use each protractor to determine the measure of the angle. 0.. L G M S B I Lesson ~ Measuring And Naming Angles 5

6 use a protractor to measure each angle to the nearest degree D M G R U E F A N P 3 V I use a protractor to draw each angle. Label the angle(s). 8. m SAM = m YAK = 5 0. m CAT = 67. an 80 angle with PQ and PR. two 35 angles with the same vertex 3. two adjacent angles that are 50 and 00 use the diagram below to name an angle with the specified measure D C E M B A G 6 Lesson ~ Measuring And Naming Angles

7 tic-tac-toe ~ duplicating Angles Constructions are part of geometry. A geometric construction is made by using a compass and straightedge. Follow the steps to duplicate ABC on a piece of notebook paper. A step : Trace ABC on your paper. step : Use a straightedge to draw a ray. This will be one side of the duplicate angle. step 3: Place the stylus or sharp point of a compass on the vertex of the traced angle. Draw an arc on the angle. B Place stylus here. C step 4: Without changing the setting on the compass, place the stylus on the endpoint of the duplicate ray and draw an arc. step 5: Use the compass to measure the width of the arc drawn on the original angle. Place the stylus on the intersection of a side and the arc. Adjust the compass so the pencil is touching the other intersection point. Place stylus here. step 6: Without changing the setting on the compass, place the stylus on the intersection of the arc and duplicate ray. Make a small arc intersecting the larger arc. step 7: Use a straightedge to connect the endpoint of the duplicate ray. This is the second ray needed to complete the duplication of ABC.. Use a protractor to draw a 60 angle.. Using only a compass and straightedge, duplicate the 60 angle. 3. Measure the duplicated angle with a protractor to check accuracy. 4. Repeat these steps on a 5, 8 and a 60 angle. 5. Which step is the most difficult for you in this process? Why? Lesson ~ Measuring And Naming Angles 7

8 classifying angles Lesson In Lesson you named and measured angles. Angles can be classified into groups by their degree measure. explore! classify an angle step : Use a protractor to measure the angles in each group. Record each measurement. group A group B group C step : Answer each question for each group. a. How are the angles in each group alike? b. What do you notice about the degree measures of the angles in each group? step 3: Write at least two sentences describing each group of angles. Include information about their degree measures. step 4: Compare what you wrote in step 3 to the definitions given below. Identify which definition is appropriate for each group. Acute angle an angle that measures more than 0 and less than 90 Right angle an angle that measures exactly 90 Obtuse angle an angle that measures more than 90 but less than 80 8 Lesson ~ Classifying Angles

9 There are four classifications of angles based on their degree measure. In the Explore!, you learned about acute, obtuse and right angles. An angle that has a measure of 80 is called a straight angle. right Angle straight Angle Right angles are often identified by drawing a small square in the vertex of the angle. If a square is present in the vertex of an angle, the angle measures 90⁰. example Classify each angle by measuring it with a protractor. a. C b. J A T c. O F d. I L X D O G solutions a. m CAT = 64 so the angle is acute. b. m JIL = 0 so the angle is obtuse. c. The box drawn at the vertex shows that m FOX = 90. d. m DOG = 80 so it is a straight angle. Angles with equal measures are congruent. The symbol for congruent is. Congruent angles are identified in diagrams with congruence marks. The two marks on the arc inside each angle are congruence marks. They show RED BLU. This is read, Angle RED is congruent to angle BLU. D X is congruent to Y because each measures 90. Each angle is E B a right angle as indicated by the square in each vertex. L X U R Y Lesson ~ Classifying Angles 9

10 example sketch a diagram of congruent and adjacent angles. solution The angles share OC which makes them adjacent. The congruence marks indicate the angles are congruent. D C O B example 3 use the information in the diagram to write an equation. solve for x. S P I (5x ) M E 39 L solution Congruence marks show the angles are congruent. SIM PLE Write the equation. 5x = 39 Add to both sides of the equation. + + Divide both sides of the equation by 5. 5x 5 = 50 5 x = 0 Check the solution by substituting 0 for x. 5x = 39 5(0)? = 39 50? = = 39 example 4 solution JAK is congruent to hil. The measure of JAK = ( 3x) and the measure of hil = (44 x). solve for x. Then find the degree measure of each angle. Write an equation showing the angles JAK HIL have equal measures. 3x = 44 x Add x to each side of the equation. +x +x x = 44 Subtract from each side of the equation. Divide each side of the equation by. x = 3 x = 6 Write the given expression for each angle. m JAK = 3x m HIL = 44 x Substitute 6 for x. = 3( 6) = 44 ( 6) Simplify. = + 48 = Add. = 60 = 60 m JAK = 60 and m HIL = 60. Both angles are equal which verifies that they are congruent. 0 Lesson ~ Classifying Angles

11 exercises estimate the degree measure for each angle. Classify each angle as acute, obtuse, right or straight sketch a diagram for each description. Label each angle. 7. two congruent angles 8. GUY is obtuse 9. GAL is acute 0. JKL POM. a right angle that can be identified using 3 names. two adjacent, right angles 3. two congruent angles that share a vertex 4. PQR is straight use each diagram to solve for x m PQR = (x + ) (3x ) R P Q (x 0) ROY MAN 0. m XYZ = m ABC (4x + 6) m XYZ = (5 + x) R M m ABC = (3x + ) 30 O (7x + 3). JAM and GEM are congruent angles. The measure of JAM is (8x + 5) and m GEM = (x + 75). a. Solve for x. b. Find the measure of each angle.. SML is an acute angle. The measure of SML = (x 7). What must x be less than? Y A N (5x + 3) Lesson ~ Classifying Angles

12 3. m LRG = (4x + ) a. If LRG is a right angle, what must x equal? b. If LRG is an acute angle, what must x be less than? c. If LRG is an obtuse angle, what must x be greater than? review sketch a diagram to represent each figure. 4. RMP 5. PIE also called and 6 which are adjacent use a protractor to measure each angle use a protractor to draw an angle with the given measure tic-tac-toe ~ ProtrActor guide step : Create a user s guide describing how to measure and draw angles with a protractor. Your teacher may use the guide to refresh a substitute teacher or for students who are absent the day of the lesson. step : Create a worksheet that can be completed using the guide. step 3: Make an answer key for the worksheet. Lesson ~ Classifying Angles

13 complementary and supplementary angles Lesson 3 Individual angles are classified as acute, right, obtuse or straight. Special pairs of angles can also be classified. In Lessons 3-6, special pairs of angles and their relationships will be examined. explore! complementary vs supplementary step : Look at the angles in the chart. a. What similarities do you notice about the pairs of angles called supplementary angles? b. What similarities do you notice about the pairs of angles called complementary angles? supplementary AngLes CoMPLeMentArY AngLes and are supplementary. R O M E ROM and MOE are complementary. C 40 A T D 40 O G CAT and DOG are supplementary. C 70 A R W 0 K L CAR and WLK are complementary. L E F T G R LEF and RGT are supplementary. and are complementary. m 6 = 80 and m 7 = 00 6 and 7 are supplementary. m 8 = 45 and m 9 = 45 8 and 9 are complementary. step : Write a definition for supplementary angles. step 3: Write a definition for complementary angles. step 4: Give at least two examples of angle measures for each type of angle pair listed below. a. Supplementary angles b. Complementary angles Lesson 3 ~ Complementary And Supplementary Angles 3

14 Complementary and supplementary angles are special pairs of angles. Complementary angles are two angles with a sum of 90. Two angles with a sum of 80 are called supplementary angles. These special pairs of angles may or may not be adjacent. example use the diagram to find m PAr. r solution m PAR and m TAR are supplementary. 45 P A t Supplementary angles have a sum of 80. m PAR + 45 = 80 Subtract 45 from both sides of the equation m PAR = 35 m PAR is 35 Check the solution ? = = 80 example gra and Ins are supplementary. a. Write an equation to solve for x. b. determine the measure of each angle. G R (x + 4) A I (3x + ) N S solutions a. Supplementary angles have a sum of 80. m GRA + m INS = 80 Write an equation. (x + 4) + (3x + ) = 80 Combine like terms. 5x + 5 = 80 Subtract 5 from each side. 5 5 Divide each side by 5. 5x 5 = 75 5 x = 35 b. Write the given expression for each angle. m GRA = (x + 4) m INS = (3x + ) Substitute 35 for x. = ((35) + 4) = (3(35) + ) Multiply. = (70 + 4) = (05 + ) Add. = 74 = 06 m GRA + m INS = ? = = 80 The measure of GRA is 74. The measure of INS is Lesson 3 ~ Complementary And Supplementary Angles

15 example 3 use the diagram to write an equation. solve for x. o 6 e (x + 5) M h solution Complementary angles have a sum of 90. m HOM + m MOE = 90 Substitute the degreee measures. 6 + (x + 5) = 90 Combine the like terms. x + 67 = 90 Subtract 67 from each side of the equation x = 3 Check the solution. 6 + (3 + 5)? = ? = = 90 The value of x is 3. example 4 and are complementary angles. The measure of = (3x + 4) and m = (x + 6). a. draw a diagram. b. Write an equation and solve for x. c. Find and. solutions a. (3x + 4) (x + 6) or (3x + 4) (x + 6) b. Complementary angles have a sum of 90. m + m = 90 Substitute the degree measures. (3x + 4) + (x + 6) = 90 Combine like terms on the same side of the equation. 4x + 0 = 90 Subtract 0 from each side of the equation. 0 0 Divide by 4 on each side of the equation. 4x 4 = 80 4 x = 0 c. Write the given expression for each angle. m = (3x + 4) m = (x + 6) Substitute 0 for x. = (3(0) + 4) = (0 + 6) Multiply. = (60 + 4) Add. = 64 = 6 Check the solution ? = = 90 m = 64 and m = 6. Lesson 3 ~ Complementary And Supplementary Angles 5

16 exercises Identify each pair of angles as complementary, supplementary or neither m and m sum to A and M have a sum of 90. Write an equation for each description. solve for x. Check your solution. 9. A and B are complementary A B x x (4 + x) x x (7x 5) (x + 3) x x 5. MAN and MAP are supplementary. The measure of MAN is 57. What is the measure of MAP? 6. 5 and 7 are complementary angles. Find the measure of 7 if m 5 = The complement of Q is 3. Find m Q. 8. The supplement of U is 6. What is m U? 6 Lesson 3 ~ Complementary And Supplementary Angles

17 Find the measure of each angle in exercises 9-3. Check your solution. 9. and are supplementary; m = 3x and m = 3x 0. W And C are complementary; m W = (47 + 3x) and m C = (0 + 8x). G and H are supplementary; m G = (x + 4) and m H = (4x + ). V and W are supplementary; V is ( + 3x) and m W = (33 + x) 3. and are complementary; m = 4x and m = (x + 8) Use the figure on the right. a. Name an obtuse angle. b. Name three pairs of supplementary angles. c. Name a pair of complementary angles. d. Give possible measures for 4 and. e. Determine m review draw and label a diagram to represent each statement. 5. PAM is acute 6. BIG is obtuse 7. RHT is right use a protractor to measure each angle to the nearest degree tic-tac-toe ~ Angle Art step : Research how angles and lines are used in art. Write at least a one page summary of your findings. step : Create a work of art using the different types of angles in Block. Use an inch by 8 inch piece of paper for your work of art. Give your creation a title and sign it. Lesson 3 ~ Complementary And Supplementary Angles 7

18 vertical angles Lesson 4 Complementary and supplementary angles are types of special angles. Another pair of special angles are vertical angles. Vertical angles are formed by two intersecting lines. They have a common vertex but are not adjacent. 3 4 Two adjacent angles whose non-common sides are opposite rays are a linear pair. In the diagram above, there are four sets of linear pairs. For example, and 3 form a linear pair. If two angles form a linear pair, they are supplementary. explore! step : Trace 3 above onto a sheet of paper. the vertical angle relationship step : Place the traced 3 on top of 4. What do you notice? step 3: Repeat steps and with and. What do you find? step 4: Draw two intersecting lines on a piece of paper. step 5: Label the angles that are formed with the numbers 5, 6, 7 and 8. Identify the vertical angles in your drawing. step 6: Measure each angle in your drawing with a protractor. step 7: What can you conclude about the measure of any pair of vertical angles? Lesson 4 ~ Vertical Angles

19 Sierra and King used different methods to find the solution to the question below. Look at Sierra s and King s work. Question sierra s Work King s Work If m = 36 and m 3 = 44, what is the measure of 4? 4 3 Sierra knows that 4 and 3 are supplementary because they are a linear pair. She subtracted 44 from = 36 m 4 = 36 King knows that and 4 are vertical angles. They have the same degree measure. m 4 = 36 There is often more than one way to arrive at a correct answer. Both Sierra and King answered the question correctly but used different methods. example Find the measure of each missing angle. a. m 3 b. m c. m solutions a. Vertical angles are congruent. m = m 3 54 = m 3 b. and are a linear pair. m + m = 80 Substitute 54 for m. m + 54 = 80 Subtract 54 from each side of the equation. m = 6 c. Vertical angles are congruent. m = m 4 6 = m 4 In geometry, sketches are used as a visual representation for the information given. Sketches are not always accurate in terms of actual length or degree measure. Information given in a diagram should be used to solve a problem rather than measuring the actual lengths and degree measures with a ruler and protractor. For example, the sketch below shows that the angle is 45 and the length of the segment is 5 feet. The angle in the sketch may not measure exactly 45 and the line does not measure 5 feet. You should still use this information to solve any problem asked about the diagram feet Lesson 4 ~ Vertical Angles 9

20 example use the diagram at right. a. solve for x. b. Find the measure of each angle. (3x + 7) (x + 30) solutions a. Vertical angles have equal measures. 3x + 7 = x + 30 Subtract x from each side of the equation. x + 7 = 30 Subtract 7 from each side of the equation. 7 7 x = 3 Divide each side of the equation by. x =.5 b. Write the given expression (3x + 7) (x + 30) for each angle. Substitute the solution for x. (3(.5) + 7) ( ) Multiply Add The measure of each angle is 4.5. Each special angle pair has properties that are important to remember. The sum of complementary angles is 90. The sum of supplementary angles or linear pairs is 80. Vertical angles are congruent. exercises use the diagram at right. determine the measure of each unknown angle.. If m = 50, find the following:. If m 4 = 53, find the following: a. m =? a. m =? b. m 3 =? b. m =? c. m 4 =? c. m 3 =? 3. If m 3 = 6, find the following: 4. If m = 6, find the following: a. m =? a. m =? b. m =? b. m 3 =? c. m 4 =? c. m 4 =? 4 3 sketch a diagram to represent each situation. Label each diagram and 8 are vertical angles and 0 are obtuse vertical angles. 7. and are a linear pair. 8. ABC and DBE are vertical is an acute angle. and complementary angles. 0 Lesson 4 ~ Vertical Angles

21 Identify each special angle pair as vertical angles or a linear pair. solve for x. Check your solution (5x + 4) (4x + 30) (8x + ) (6x + 4) (4x 4) (x + ) (3x + 7) 6x (5x + 9) 89 (3x + 0) 6 (8x + 3) (7x + ) 8. Use the diagram at the right. a. Solve for x. b. Find m ABC. c. Find m CBD. (4x + 7) C (x ) A B D 9. Use the diagram at the right. a. Solve for x. b. Find m WX Y. c. Find m TXY. (6x + 3) U T X Y (0x 7) W Lesson 4 ~ Vertical Angles

22 review Match each diagram to a description from the word bank. some diagrams may match more than one description. Acute angle Obtuse angle Right angle Congruent angles Straight angle Vertical angles Supplementary angles Linear pair Complementary angles K 40 I S 3. m PQR = 0 4. m ABC = P Q R tic-tac-toe ~ BooK of Poetry step : Write a separate poem about each special angle pair: linear pair, supplementary, complementary, vertical, alternate interior, alternate exterior, corresponding and same-side interior angles. Title each poem with the name of the special angle pair. step : Put the eight poems in a booklet. Include diagrams and illustrations. Lesson 4 ~ Vertical Angles

23 alternate exterior and interior angles Lesson 5 The words interior and exterior are used in many non-math situations. The word interior is used to describe things that are inside. Things that are exterior are on the outside. The two special angle pairs in this lesson are called alternate interior angles and alternate exterior angles. Look at the diagrams in the table below. What does it mean for a pair of angles to be alternate exterior angles? What about alternate interior angles? Alternate exterior Angles Alternate Interior Angles and 8 are alternate exterior angles. and are alternate interior angles and 7 are alternate exterior angles. 3 and 4 are alternate interior angles and 8 are alternate exterior angles. 6 and 9 are alternate exterior angles. 0 and 3 are alternate interior angles. and 4 are alternate interior angles A transversal is a line that intersects two or more lines. Alternate exterior angles are two angles on the outside of two lines and on opposite sides of a transversal. Alternate interior angles are two angles on the inside of two lines and on opposite sides of a transversal. X B 4 3 A In this diagram, XY is the transversal. P Q AB PQ reads, Line AB is parallel to line PQ. Y Lesson 5 ~ Alternate Exterior And Interior Angles 3

24 explore! alternate exterior and alternate interior step : Draw two parallel lines intersected by a transversal. Make the diagram large enough to easily measure the angles with a protractor. Label your angles through 8. step : Use a protractor to measure the eight angles on the drawing from step. Record their measures. step 3: Which pairs of angles in your diagram are alternate exterior angles? What do you notice about the degree measure of each pair of alternate exterior angles? step 4: Which pairs of angles are alternate interior angles? What do you notice about the degree measure of each pair of alternate interior angles? step 5: Trace the figure from the bottom of page 3 onto a piece of paper. step 6: Slide your tracing until angles,, 3 and 4 are on top of angles 5, 6, 7 and 8. a. What do you observe? Explain. b. Does this confirm your conclusion in steps 3 and 4? step 7: Draw two lines that are not parallel. Draw a transversal that intersects both lines. step 8: Use a protractor to measure the eight angles created in step 7. step 9: Write a conclusion about the alternate interior angles and alternate exterior angles formed by parallel lines compared to those formed by non-parallel lines. example name the angle relationship between and. determine whether and are congruent. a. b. c. < < << << solutions a. Alternate interior. The angles are congruent because the lines are parallel. b. Alternate exterior. The angles are congruent because the lines are parallel. c. Alternate exterior. The lines are not parallel, so the angles are not congruent. 4 Lesson 5 ~ Alternate Exterior And Interior Angles

25 << example Identify the special angle pair relationship. solve for x. << (8x + 4) 86 solution The angles are alternate exterior angles. l m Lines l and m are parallel so the alternate exterior angles are congruent. 8x + 4 = 86 Subtract 4 from each side of the equation. 4 4 Divide each side of the equation by 8. 8x 8 = 8 8 x = 0.5 Check the solution. (8x + 4) = 86 Substitute 0.5 for x. (8(0.5) + 4) =? 86 Add =? = 86 example 3 use the figure at right. a. solve for x. b. Find the measure of the angles. (x + 5) (9x 58) solutions a. The lines are parallel so alternate interior angles are congruent. 9x 58 = x + 5 Subtract x from each side of the equation. x x Add 58 to each side of the equation. 7x 58 = Divide each side of the equation by 7. 7x 7 = 63 7 x = 9 b. Write the given expression for each angle. (9x 58) (x + 5) Substitute 9 for x found in part a. (9(9) 58) ((9) + 5) Multiply. (8 58) (8 + 5) Simplify. 3 3 Each angle measures 3. Alternate interior angles are congruent so part b verifies the solution for x. Lesson 5 ~ Alternate Exterior And Interior Angles 5

26 exercises use one of the following special angle pairs to identify the relationship of the angles shown. Alternate exterior Alternate Interior Vertical Linear Pair y y 90 name the special angle pair relationship between and. explain whether name the special angle pair relationship. solve for x > (4x + 3) > 70 0 (3x + 0) > 03 > 8x 6 Lesson 5 ~ Alternate Exterior And Interior Angles

27 solve for x. Then find the measure of each identified angle. Check your solution (6x + 0) (x + 00) (x + ) (3x ) (6x 7) (5x + 0) and are vertical angles. 5 and 8 are m = (5x + 7) supplementary m = (3x + 5) m 5 = (3x 40) m 8 = (7x 0) x 56. Explain how to distinguish between alternate exterior angles and alternate interior angles. review Write two possible names for each angle P W 5 B A 3 M o K sketch and label a diagram for each description. 6. an acute angle 7. vertical angles with each angle measuring adjacent supplementary angles 9. complementary angles that are not adjacent tic-tac-toe ~ transversal collage step : Find at least 0 photographs or pictures of transversals. You can take photographs, locate and print pictures from the internet or cut pictures from magazines or newspapers. step : Identify the transversal and special angle pairs in each picture. step 3: Make a collage to display the pictures. Lesson 5 ~ Alternate Exterior And Interior Angles 7

28 tic-tac-toe ~ Bisecting Angles To bisect an angle means to cut it in half. You can bisect an angle using two different methods. One construction method to bisect an angle uses patty paper. Another method uses a compass and straightedge. Bisecting an Angle using Patty Paper or tracing Paper step : Draw or trace an angle onto a piece of patty paper or tracing paper. Record the measure of the angle. step : Fold one ray of the angle onto the other ray of the angle. step 3: Trace the crease and measure each angle. Bisecting an Angle using A Compass and straightedge step : Draw or trace an angle. Record the measure of the angle. step : Place the stylus or sharp point of a compass on the vertex. Use the compass to draw an arc through the angle. step 3: Place the stylus on one of the points of intersection between a ray of the angle and the arc from step. Draw an arc as shown. Place stylus here. step 4: Repeat step 3 at the other point of intersection. step 5: Use a straightedge to draw a ray from the vertex to the intersection of the two arcs drawn in steps 3 and 4. step 6: Measure each angle.. Use a protractor and patty/tracing paper to draw 90, 4, 5 and 60 angles. Each angle should be on a separate sheet of patty/tracing paper.. Bisect each angle by folding. 3. Use a protractor to draw another set of angles with the measures listed in # on regular paper. 4. Bisect each angle using a compass and straightedge. 5. Lay your matching patty/tracing paper constructions on top of each compass construction. 6. Summarize each method of construction. Include which method you prefer and explain why. Discuss the pros and cons of each method. 8 Lesson 5 ~ Alternate Exterior And Interior Angles

29 corresponding and same - side interior angles Lesson 6 Two additional special pairs of angles formed by two lines and a transversal are corresponding angles and same-side interior angles. Corresponding angles are two angles on the same side of a transversal. One angle is an exterior angle and the other angle an interior angle. They are not adjacent angles. Same-side interior angles are between the two lines on the same side of a transversal explore! more angle pairs step : If two parallel lines are cut by a transversal, predict: a. What is the relationship between a pair of same-side interior angles? b. What is the relationship between a pair of corresponding angles? step : Use a straight edge to draw a pair of parallel lines with a transversal. Make the figure large enough that you can measure the angles with a protractor. step 3: Use a protractor to measure each of the eight angles. Record each measure. step 4: What do you observe about the corresponding angle measures? step 5: Look at the measures of the same-side interior angles. What do you observe about their measures? step 6: Must the lines be parallel for these relationships to occur? Draw a figure without parallel lines to test your answer. Lesson 6 ~ Corresponding And Same - Side Interior Angles 9

30 example name the special angle pair relationship between and. a. b. c. solutions a. corresponding angles b. same-side interior angles c. corresponding angles Claudia and Ping come to different conclusions when finding the missing angles in the diagram below. Determine who is correct and why Claudia s Work m = 7 m = 08 m 3 = 08 m 4 = 7 m 5 = 7 m 6 = 08 m 7 = 7 Ping s Work m = 7 m = 08 m 3 = cannot determine m 4 = cannot determine m 5 = cannot determine m 6 = cannot determine m 7 = 7 Two lines in the diagram are intersected by a transversal. It is not known whether the lines are parallel. Unless lines are marked parallel it cannot be assumed they are parallel. Ping is correct. The measures of 3, 4, 5 and 6 cannot be determined. 30 Lesson 6 ~ Corresponding And Same - Side Interior Angles

31 example Write an equation and solve for x. (4x + 9) 7 solution Corresponding angles have equal measures. 4x + 9 = 7 Subtract 9 from each side of the equation. 9 9 Divide each side of the equation by 4. 4x 4 = 08 4 x = 7 Check the solution. Substitute 7 for x in the equation. 4(7) + 9? = ? = 7 7 = 7 example 3 Write an equation and solve for x. Then find the measure of each identified angle. << (3x + ) (3x + 44) << solution The lines are parallel so the same side interior angles sum to 80. (3x + ) + (3x + 44) = 80 Combine like terms. 6x + 45 = 80 Subtract 45 from each side of the equation Divide each side of the equation by 6. 6x 6 = 35 6 x =.5 Find the measure of each angle. (3x + ) (3x + 44) Substitute.5 for x. (3(.5) + ) (3(.5) + 44) ( ) ( ) =? = 80 Lesson 6 ~ Corresponding And Same - Side Interior Angles 3

32 exercises name the special angle pair relationship between and in each diagram << << > > << << << << << << 0. Which diagrams in exercises -9 have parallel lines? solve for x (47 + x) > > 54 4x 9 (5x + 34) solve for x. Then find the measure of each identified angle. Check your solution (x + 6) 0 46 (4x + 40) (90 5x) (6 x) 3 Lesson 6 ~ Corresponding And Same - Side Interior Angles

33 Find the measure of each numbered angle. Check your solution (0x + 5) 3 (8x + 5) (4x + ) (6x 7) sketch a diagram to represent each special pair of angles. Label the angles in each pair as and. 0. congruent same-side interior angles. alternate interior angles that are not congruent. acute corresponding angles 3. right alternate exterior angles 4. obtuse same-side interior angles 5. acute vertical angles review use a protractor and/or straightedge to draw each diagram. Label each diagram. 6. m GUM = 6 7. m RPQ = parallel lines with transversal AB 9. right angle named name each special angle pair. solve for x x 9 3x x (5x + ) (3x + 44) tic-tac-toe ~ FliP BooK Special Angle Pairs step : Create a flip book showing all of the special angle pairs in Block. Supplementary angles Vertical angles Corresponding angles Alternate interior angles Same-side interior angles Alternate exterior angles Complementary angles Linear pairs step : Include a definition, a diagram and an example for each special angle pair. Lesson 6 ~ Corresponding And Same - Side Interior Angles 33

34 tic-tac-toe ~ PuZZling Angles Find the measure of the numbered angles in each diagram.you will need to research about the sum of the angles in a triangle before completing the puzzle tic-tac-toe ~ crossword Create a crossword puzzle using all of the vocabulary from Block. Make a blank master copy and an answer key. 34 Lesson 6 ~ Corresponding And Same - Side Interior Angles

35 review BLoCK vocabulary acute angle corresponding angles right angle adjacent angles degree same-side interior angles alternate exterior angles linear pair straight angle alternate interior angles obtuse angle supplementary angles angle protractor transversal complementary angles ray vertex congruent vertical angles Lesson ~ Measuring and Naming Angles use a protractor to measure each angle to the nearest degree List the four names for each angle. 4. O 5. 5 M P X 5 Y sketch and label a diagram to represent each angle. use a protractor, as needed. 6. m YOU = 5 7. m BAT = HAM and HAP are adjacent Z Block ~ Review 35

36 Lesson ~ Classifying Angles Classify each angle as acute, right, obtuse or straight sketch and label a diagram.. CAT DOG 3. a right angle that can be named three different ways set up an equation and solve for x. Check your solution (35 + x ) 3 (x + 6 ) (3x ) 7. m BIG = (x + 4) a. What must x be equal to if BIG is a right angle? b. What must x be less than if BIG is acute? c. What must x be greater than if BIG is obtuse? Lesson 3 ~ Complementary and Supplementary Angles Identify each pair of angles as complementary, supplementary or neither solve for x. Check your solution... m A = (5 + x) 3. m P = (0 + 3x) A and P are complementary. 4 4 x (x + 7) 36 Block ~ Review

37 sketch a diagram for each situation. Label it and solve for x. 4. PAR and TYE are supplementary; m PAR is 83 and m TYE is (x + 5) 5. F and G are supplementary; m F is 46 and m G is (3x 5) 6. and are complementary; m is x and m is 3x Lesson 4 ~ Vertical Angles determine the measure of the angles labeled a, b and c b a c 45 a b 3 c name the special angle pair. solve for x. Check your solution (x + 8) 4 (x + ) (5x + 7) (6x + 7) (4x + 0) (5x) (5 + 3x) (5x) (0 + 7x) (x + 76) Lesson 5 ~ Alternate Exterior and Interior Angles name the special angle pair relationship between and. explain whether or not the angles are congruent Block ~ Review 37

38 solve for x. Then find the measure of each angle. Check your solution (x + ) (3x 35) (60 x) (8 + x) (5x + ) (3x + 40) Lesson 6 ~ Corresponding and Same-Side Interior Angles name the special angle pair relationship between and Copy the table below. List each special angle pair from the blue box in one category. Assume lines are parallel for *angle pairs. Linear Pairs *Alternate exterior Angles *Alternate Interior Angles *Corresponding Angles Vertical Angles Complementary Angles *same-side Interior Angles supplementary Angles have a sum of 90 have a sum of 80 Are equal in measure Write an equation to solve for x. Then find the measure of each angle. Check your solution (x + 0) (x + 5) (3x + 70) (7x + 30) (8x ) (48 x) 38 Block ~ Review

39 Jerry nursery manager turner, oregon CAreer FoCus I am a nursery manager. I work with a staff planning and planting 5 different tree species. Some of the seedlings I plant are very recognizable. I plant Douglas Fir, Noble Fir, Ponderosa Pine and Giant Sequoia. Some of the trees I grow are for planting around homes. Others go into the woods for reforestation. When loggers cut down trees, the trees I grow are planted in their place. This helps make sure that there will be trees to harvest in the future. I help grow 6.5 to 7.5 million trees for about 50 different customers. I work at a small nursery, but some nurseries in the Northwest grow over 5 million trees a year. I use math every day in my job. When preparing to plant, I have to know how many trees will fit on each acre. Some seedlings are planted at 75 per foot and some are planted at 8 per foot. Math helps me to determine how many seeds I will need to make sure I will have enough trees to fill a customer s order. Fertilizers are another area where I use math. I take soil samples and calculate at what rate I should apply fertilizer to make sure that the seedlings grow well. When I prepare to ship trees to a customer, I also have to use math to make sure that they are counted correctly and packaged right for each customer. Most nursery managers have a Bachelor of Science degree in Agriculture or a related horticultural field. These degrees require high-level math classes like calculus. About 0% of nursery managers do not have a college degree. Those managers usually work their way up to a manager position with many years of work and experience. Nursery manager salaries start at around $35,000 to $40,000 per year and can get as high as $65,000 or more per year. Often the pay is related to the size of the nursery and how many trees the nursery sells. I feel lucky to be doing something I enjoy and making a positive contribution to the world I live in by growing trees. I have helped grow over 80,000,000 trees in my career. I enjoy going to the woods and seeing stands of trees I grew. I know that some of those trees will still be living when my grandchildren s grandchildren are old. Block ~ Review 39