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Transcript
Geometric Figures Chapter Focus You will identify and describe basic geometric figures. Angle pairs such as adjacent angles and vertical angles will be used to solve problems. You will draw shapes that satisfy given conditions, and you will use transformations to produce congruent figures on a coordinate plane. ('~~~'MON Chapter at a Glance : Lesson Standards for Mathematical Content 8-1 Building Blocks of Geometry PREP FOR PREP FOR 8-2 Classifying Angles CC.7.G.5 8-3 Line and Angle Relationships CC.7.G.5 8-4 Angles in Polygons CC.7.G.2 8-5 Congruent Figures PREVIEW Of Problem Solving Connections Performance Task Assessment Readiness CORE "~"" .......... CC.7.G.5, CC.7.G.6 CC.S.G.2 Unpacking the Standards Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter. What It Means to You You will learn about supplementary, complementary, vertical, and adjacent angles. You will solve simple equations to find the measure of an unknown angle in a figure. EXAMPLE II line p. Find the measure of each angle. Line n p n Adjacent angles formed by two intersecting lines are supplementary. mL5 + 55° 1800 mL5:= 1800 - 55° Vertical angles are congruent. mL6:= 55° mL7 = 1250 Alternate interior angles are congruent. mL3 == 55 0 Alternate exterior angles are congruent. mL2 1250 Corresponding angles are congruent. mL1 55 0 mL4 = 125 0 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Use geometry software to construct an obtuse triangle with angles that measure 120°, 26°, and 34°. protractor (transportador) A tool for measuring angles. Use the" dilate" tool to enlarge and reduce the triangle. What can you conclude about constructing triangles with given angles of 120°, 26°, and 34°? You can draw several triangles with various side lengths with angles that measure 120°, 26°, and 34°. Key Vocabulary adjacent angles (angulos adyacentes) Angles in the same plane that have a common vertex and a common side. complementary angles (angulos complementarios) Two angles whose measures add to 90°. congruent angles (angu/os congruentes) Angles that have the same measure. line (linea) A straight path that has no thickness and extends forever. line segment (segmento de recta) A part of a line made of two endpoints and all points between them. plane (plano) A flat surface that has no thickness and extends forever. point (punto) An exact location that has no size. ray (rayo) A part of a line that starts at one endpoint and extends forever in one direction. reflection (reflexi6n) A transformation of a figure that flips the figure across a line. rotation (rotaci6n) A transformation in which a figure is turned around a point. supplementary angles (angulos suplementarios) Two angles whose measures have a sum of 180°. translation (traslaci6n) A movement (slide) of a figure along a straight line. vertical angles (angulos opuestos por el vertice) A pair of opposite congruent angles formed by intersecting lines. Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _ii:!'< Building Blocks of Geometry Essential question: How do you identify and describe basic geometric figures? A~ln! is an exact location in space with no size. A fift~ is a straight path with no thickness that extends forever in opposite directions. point P ~ LM A D~~i~ is a flat surface with no thickness that extends forever. plane ABC Ati:, is a part of a line. It has one --> endpoint and extends forever in one direction. RS A~~p~i~~ijl is a part of a line or ray .p M ~ s ~ H GH between two endpoints. ~ Drawing Geometric Figures Use a ruler or straightedge to help you sketch each geometric figure in the drawing box. ~ co Q. A Draw line XY. E 8 O'l c: ~ ::a::J 0 1:: ::J ~ co :c .S: Draw ray XZ so that point Z does not lie on lineXY. The endpoint is always listed fIrst in the name of a ray, so the endpoint of ray XZ is point _ _ __ iE ~ c: E Draw segment YZ. O'l ::J o :c @ The endpoints of segment yz are and _ _ __ points Draw plane XYZ. Write a set of directions for drawing at least 3 geometric figures like those described above. Trade your directions with a partner, and draw the figures he or she describes. Then check each oth~r's work. Chapter 8 311 Lesson 1 • • • • Point: Use a capital letter. Line: Use any two points on the line or a lowercase letter. Plane: Use any three points in the plane that are not on the same line . Ray: Use the endpoint and another point on the ray. List the endpoint first . • Line segment: Use the endpoints. Naming Geometric Figures i~ A List all possible names of the line shown. You can name a line by using _ _ _ points. The order in which you list the points does I does not matter. Possible names for the line are fiB, ___ _____ ,and _____ .•~.. List all possible names of the ray with endpoint D shown. You can name a ray by using ____ points, one ofwhich must be the endpoint. The endpoint of the ray must be listed first I second. Possible names for the ray are and ______ List all possible names of each figure. 2a. the plane 2b. the segment with endpoints K and M ____ and ______ K ~M 2c. How is a line segment similar to a ray? How are they different? m. What mistake Error Analysis Christi named the line shown as did she make? What is a correct name for the line? Chapter 8 312 T ~ Lesson 1 Name ________________________ Identify the figures in the diagram. 1. three points __________________ 2. one line _ _ _ _ _ _ _ _ _ __ 3. a plane _______________ 4. four rays ________________ 5. three line segments ____________ Identify the figures in the diagram. 6. four points _ _ _ _ _ _ _ __ 7. three lines ______________ 8. a plane _____________ 9. three rays _______________ 10. four line segments __________________ > C III a. E o u en c ~ Identify the figures in the diagram. 11. four points __________ :0 ::I "' t:: ::I o 12. two lines ___________ ~ III I .~ !j;; ~ C o 1: en ::I o 13. a plane ________________ 14. four rays ____________ I @ 15. five line segments ______________ 16. Identify the line segments that are congruent in the figure. -l++_.....,c A,---+f---"i'--__ F Chapter 8 313 Practice and Problem Solving Write the correct answer. The drawing shows a section of the Golden Gate Bridge in San Francisco. 1. Identify two lines that are suggested by the bridge. 2. Identify a ray and a line segment that are suggested by the bridge. 3. Identify two lines in the figure that are in the same plane. 4. Identify a plane in the figure. Choose the letter for the best answer. The drawing is an artist's sketch for an abstract painting. r-t--t=---t=-H-t----lF 7. Which line segment is congruent to 8. Which line segment is congruent to EO? OF? A AC C OF F EO H CJ B CD o G HM J FM Chapter 8 GH 314 Practice and Problem Solving Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date----mw Classifying Angles Essential question: How can you use angle pairs to solve problems? Recall that two rays with a common endpoint form an angle. The two rays form the sides of the angle I and the common endpoint marks the vertex. You can name an angle several ways: by its vertexl by a point on each ray and the vertexl or by a number. B c Angle names: LABCI LCBA I LBI Ll It is useful to work with pairs of angles and to understand how pairs of angles relate to each other. Congruentangles are angles that have the same measure . • =rr:CI!'", •• J%. .... B Measuring Angles Using a rulerl draw a pair of intersecting lines. Label each angle from 1 to 4. Use a protractor to help you complete the chart. .,.> M~aslJ..ef)f.AJlgle· Angle mL1 mL2 mL3 mL4 mL1 +mL2 mL2 + mL3 mL3 + mL4 mL4+ mL1 .•. Conjedure Share your results with other students. Make a conjecture about pairs ofangles that are next to each other. Chapter 8 315 Lesson 2 ~~i~~~~~i~~ are pairs of angles that share a vertex and one side but do not overlap. ~~~==~~:~~~ are two angles whose measures have a sum of 90°. ~ ~;~_l~are two angles whose measures have a sum of 180°. You have discovered in Explore 1 that adjacent angles formed by two intersecting lines are supplementary. t:••~~B;M:~I~]:"0\, Identifying Angles and Angle Pairs Use the diagram below. ..14: Name a right angle. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ iii' Name a pair of adjacent angles. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Name a pair of complementary angles. _ _ _ _ _ _ _ _ _ _ _ _ _- - - - - - Name an angle that is supplementary to LCFE. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ ~... Name an angle that is supplementary to LBFD. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ iF Name an angle that is supplementary to LCFD. _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Name a pair of non-adjacent angles that are complementary. _ _ _ _ _ _ _ _ _ __ 2a. What is the measure of LDFE? Explain how you found the measure. 2b. Are LCFB and LDFE adjacent angles? Why or why not? Are LBFD and LAFE adjacent angles? Why or why not? Chapter 8 316 Lesson 2 Wtf~~l!;le\ Finding Angle Measures Find the measure of each angle. ~; LBDC c A LBDC and _ _ _ _ are _ _ _ _ _ _ _ _ _ _ angles. The sum of their measures is _ _ _ __ Write an equation to help you find the measure of LBDC. 75 +x In the box, solve the equation for x. mLBDC= If· LEHF • • E • • G LEHF and _ _ _ _ are _ _ _ _ _ _ _ _ _ angles. The sum of their measures is _ _ _ __ In the box, write and solve an equation to help you find mLEHF. mLEHF= 3a. Find the value of x, and mL/ML. L j mL/ML=x= Chapter 8 317 N Lesson 2 PRACTICE For 1-5, use the figure. 1. mLQUP+mLPUT= _ _ _ _ _ _ _ _ _ _ __ 2. Name a pair of supplementary angles. 3. Name a pair of complimentary angles. 4. Name a pair of adjacent angles. 5. What is the measure of LQUN? Explain your answer. Solve for the indicated angle measure or variable. 6. mLYLA= _ _ __ 1. x =_______ S L~ • J A 8. The railroad tracks meet the road as shown. The town will allow a parking lot at angle] if the measure of angle] is greater than 38°. Can a parking lot be built at angle]? Why or why not? 9. A student states that when the sum of two angle measures equals 180°, the two angles are complementary. Explain why the student is incorrect. Error Analysis Chapter 8 318 Lesson 2 Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ Date _ _ __ <: Tell whether each angle is acute, right, obtuse, or straight. 1. / 2. 3. Use the diagram to tell whether the angles are complementary, supplementary or neither. 4. LAQC and LGQC 5. LBQD and LDQE 6. LCQEand LEQF ~ [ E o 7. LGQF and LFQE u Cl .:: ~ :0 ;;;J a. t:: ;;;J o 8. LBQC and LDQC u :;; :r: .5 :to ~ .:: .s ..r:: Cl ;;;J 9. Angles Wand Xare supplementary. If mLWis 3r, what is mLX? o :J: @ 10. Angles S and Tare complementary. If mLS is 64°, what is mLT? 11. Angles C and 0 are supplementary. If mLC is 83°, what is mLD? 12. Angles U and V are complementary. If mLU is 41°, what is mLV? Chapter 8 319 Practice and Problem Solving Write the correct answer. The drawing shows a scene on a calendar. 1. L1 and L2 are complementary angles. If L1 measures 35°, what is the measure of L2? 2. L3 and L4 are supplementary angles. If L3 measures 50°, what is the measure of L4? 3. Which angle is an obtuse angle: L6 or L7? 4. Which angle labeled on the drawing is a right angle? Choose the letter for the correct answer. Use the diagram to complete Exercises 5 and 6. 5. Which of the following could be the measures of LTZU and LQZR? A mL TZU 55° and mL QZR = 55° B mL TZU 25° and mL QZR = 90° C mLTZU= BO° and mLQZR= 100° D mLTZU = 35° and mLQZR = BO° s R 6. If LRZS measures 35°, what is the measure of LSZT? F 155° G 145° H 55 0 J 45° 7. LA and LB are complementary angles. The measure of LB is 4 times the measure of LA. What are the measures of the angles? A mLA 16° and mLB = 64° B mLA 1Bo and mLB = 72° C mLA:!:: 36° and mLB = 144° D mLA 45° and mLB = 1350 Chapter 8 B. The hands of a clock form an acute angle at 1:00. What type of angle do they form at 4:00? F acute G right H obtuse J straight 320 Practice and Problem Solving Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date----.iil Line and Angle Relationships Essential question: How do you use vertical angles to solve problems with figures? ~djat~:~~ have a common vertex and a common ray, but do not have any interior points in common. Vertical angles are opposite angles formed by two intersecting lines. Angles Formed by Intersecting Lines Use the figures below to determine relationships between pairs of angles formed by intersecting lines. A Use a protractor to measure the numbered angles in the figures to the nearest degree. Record the measures in the table. Figure 1 Figure 2 Figure 1 Figure 2 .11·· Add the measures of each pair of adjacent angles and record the sums in the table. Figure 1 Figure 2 '(:.. Describe the pattern in the sums of the measures of the adjacent angles of each figure. I) In each figure, Ll and L3 are vertical angles, and Land L are vertical angles. Describe the pattern in the measures of each pair of vertical angles. Chapter 8 321 Lesson 3 ! 1a. Make a Conjecture Make a conjecture about any pair of adjacent angles formed by a pair of intersecting lines. I i ,~ ~. Make a Conjecture Make a conjecture about any pair of vertical angles. PRACTICE The figure is formed by a pair of intersecting lines. Use the figure for each problem. 1. Use a protractor to fmd the measures of L1, L2, and L3. mL1 =___---', mL2 =___----', and mL3 = ____ 2. Explain how you could find the measure of L 1 without using a protractor. 3. Explain how you could find the measure of L2 without using a protractor. @ ::t: oc: '"S:r ::J s: ;:s ::J The figure shows the intersection of Pine Street, West Avenue, and Shady Lane. Use the figure for each problem. ...::t: 8 c: 4. Write and solve an equation to find the measure of L 1. "S!:c: ;:+ .,;' :r :;' -----7~~---West Ave. '"on 3 al 5. What is the measure of L2? Justify your answer without using a protractor. 6. ~ Shady Ln. L2 and L3 are complementary angles. Write and solve an equation to find the measure of L3. Chapter 8 322 Lesson 3 , I Name _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ __ ~ '"--, Find the measure of each angle. 2. L2 andL4 1. L5 and L6 ~ . 6314 4. L6 and L7 3. L4 and L8 y 3 4 87" 8 6. L3 and L4 5. L1 and L2 x Chapter 8 323 Practice and Problem Solving Write the correct answer. Use the diagram below for 1-3. 1. What is the measure of L1? 2. What is the measure of L2? 3. What is the measure of L3? Choose the letter for the best answer. The map shows the area around Falcon Park. ~~2~l 4. If 4 measures 112°, what is the measure of 2? Birch St. S. Which two angles are vertical angles? F L2 and L3 H L2 and L4 G L2 and L6 J L2 and LS Orchard St. 6. Which two angles are adjacent angles? A L1 and L4 C L3 and LS B L6 and L8 D LS and L8 7. If L4 measures 112°, what is the measure of L1? Chapter 8 324 Practice and Problem Solving Name~ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date--_',if Angles in Polygons Essential question: How can you draw shapes that satisfy given conditions? Draw each triangle with the given conditions. Triangle 1 Angles: 30° and 80° Included side: 2 inches Triangle 2 Angles: 55° and 50° Included side: 1 inch Use a ruler and a protractor to draw each triangle with the given angles and included side length. /(,. Draw Triangle 1. Step 1: Use a ruler to draw a line that is 2 inches long. This will be the included side. Step 2: Place the center of the protractor on the left end of the 2-in.1ine. Then make a 30°-angle mark. Step 3: Draw a line connecting the left side of the 2-in.line and the 30°-angle mark. This will be the 30° angle. L 2 in. Step 4: Repeat Step 2 on the right side of the triangle to construct the 80° angle. Et. Step 5: The side of the 80° angle and the side of the 30° angle will intersect. This is Triangle 1 with angles of 30° and 80° and an included side of 2 inches. Draw Triangle 2. Chapter 8 325 Lesson 4 1a. Conjedure When you are given two angle measures and the length of the included side, do you get a unique triangle? :,~:W,0"'~!''';''':,;:,x"" Co;";,:""" Two Sides and a Non-Included Angle Use a ruler, protractor, and compass to construct a triangle with given lengths of 2 inches and 1! inches and a non-included angle of 45°. A non-included angle is the angle not between the two given sides. Step 1: Use a ruler to draw a straight line. This will be part of the triangle, but does not have to measure a specific length. Step 2: As in D, place the center of the protractor on the left end of the line. Then make a mark at the correct 45-degree point. Use your ruler to make this side of the triangle 2 inches long. Step 3: Make your compass the width of I! inches. Place the sharp point on the end of the 2-inch side that you just drew in Step 2. Rotate the compass until it intersects, or meets, the bottom line twice (see figure). Step 4: The point where the compass crosses the bottom line shows where a line can be drawn that is exactly I! inches long. Use your ruler to verify the length and draw the line. 2a. LL... Is there another triangle that can be drawn with the given conditions? When you are given two side lengths and the measure of a non-included angle, do you get a unique triangle? Explain. Chapter 8 326 Lesson 4 ".I~'--'.. . "'"~~.,,,. L:; a Drawing Three Sides Use geometry software to draw a triangle whose sides have the following lengths: 2 units, 3 units, and 4 units. Step 1: Draw three line segments of 2, 3, and 4 units E of length. F c= D C =3 B A a= Step 2: Let be the base of the triangle. Place endpoint C on top of endpoint Band endpoint E on top of endpoint A. These will become two of the vertices of the triangle. D B E A a =2 C Step 3: Using the endpoints C and E as fixed vertices, rotate endpoints F and D to see if they will meet in a single point. The line segments of 2, 3, and 4 units do / do not form a triangle. zJ ~4 b=3 E A a=2 C ~&"-,_~,,,,,.,,_, ':1; _, ""'-4'<"'-~""<'''''', c~ Of B , i~!!!~!~!~~:l;j)l Repeat Steps 2 and 3, but start with a different base length. Do the line segments make the exact same triangle as the original? Use geometry software to draw a triangle with given sides of 2, 3, and 6 units. Do these line segments form a triangle? Conjecture When you are given three side lengths that form a triangle, do you get a unique triangle or more than one triangle? Chapter 8 327 Lesson 4 PRACTICE 1. On a separate piece of paper, draw a triangle that has side lengths of 3 cm and 6 cm with an included angle of 120°. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. 2. Use geometry software to determine if the given side lengths can be used to form one unique triangle, more than one triangle, or no triangle. 3. On a separate piece of paper, draw a triangle that has degrees of 30°, 60°, and 90°. Measure the side lengths. a. Can you draw another triangle with the same angles but different side lengths? b. If you are given 3 angles in one triangle, will the triangle be unique? 4. Draw a freehand sketch of a triangle with three angles that have the same measure. Explain how you made your drawing. Chapter 8 328 Lesson 4 Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ - IJOL'C _ _ __ Use a ruler, protractor, and compass to construct each figure. 1. Draw a triangle that has side lengths of 3 em and 4 em with an included angle of 90°. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. 2. Draw a triangle that has angles that measure 45°,45°, and 90°. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. Chapter 8 329 Practice and Problem Solving Use a ruler, protractor, and compass to construct each figure. 1. Draw a triangle that has sides that measure 5 em, 5 em, and 11 em. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. 2. Draw a triangle that has two angles that measure 30° and 40° with an included side length of 5 em. Determine if the given information makes a unique triangle, more than one triangle, or no triangle. Chapter 8 330 Practice and Problem Solving Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date ____ ;P1 Congruent Figures Essential question: How can you produce congruent figures on the coordinate plane? A transformation is a change in the position, shape, or size of a figure. A translation is a transformation that slides a geometric figure. A translation moves each point of a figure the same distance in the same direction. Translation [F ' ", ' ~;.•",",,",".,,''.,', ,:'/ '/ , '' ,', ., ' 'EX P LOR E\ Translating a Figure Translate !:lASe on the coordinate plane as described below. A List the coordinates of the vertices of !:lABe. ) 10 ~~-:+,~ 9 +--;".'1:'" 8 +--I---¥lIr, 7 +-"-~c'\"'f-"''"'' 8 Trace the triangle on paper and cut it out. C Place the cutout triangle on the original triangle. Translate, or slide, the cutout triangle 4 units to the right. List the coordinates of the vertices of the trans lated triangle. > c '" Co E 0 u C> 6 -r-i~;"";H\.L-r5 +--'~'!,-"""!-,,I-:;;; 4 3 2 +---1--+ 2 3 4 5 6 7 8 9 10 Compare the coordinates of the translated triangle with the coordinates of the original triangle. How do the x-coordinates compare? How do the y-coordinates compare? c ~ jj " 0" ~ J: '" 0 't .!: :E ~ c ... .s: 0 Once again, place the cutout triangle on the original triangle. This time, translate the cutout triangle 5 units down. List the coordinates of the vertices of the translated triangle. 0 ) C> "0 J: @ Compare the coordinates of the translated triangle with the coordinates of the original triangle. How do the x-coordinates compare? How do the y-coordinates compare? Chapter 8 331 Lesson 5 1a. Place the cutout triangle on the original triangle. Translate the cutout triangle 3 units to the right and then 1 unit up. List the coordinates of the vertices of the translated triangle. 1b. What If...1 What if you translated t,.ABC 2 units to the left? How would the x-coordinates of the vertices change? How would the y-coordinates of the vertices change? When you translate t.ABC, is the translated triangle congruent to the original triangle? How do you know? A is a transformation that flips a figure across ~~.itl~Ji is a transformation that turns a figure about a point. Reflecting and Rotating a Figure Reflection Rotation t I F+'i I I Y Reflect and rotate t,.JKL on the coordinate plane as described below. . • ~... list the coordinates of the vertices of t,.JKL. ~l, ) B Trace the triangle on paper and cut it out. :C Place the cutout triangle on the original triangle. Reflect, or flip, the cutout triangle across the y-axis. The arrow from vertex K shows where vertex K should be in the reflected triangle. List the coordinates of the vertices of the reflected triangle. ) Compare the coordinates of the reflected triangle with the coordinates of the original triangle. Chapter 8 332 Lesson 5 ~,; Once again, place the cutout triangle on the original triangle. This time, rotate, or turn, the cutout triangle 900 clockwise about the origin. (A 900 tum is of a full tum.) The arrow from vertex L shows where vertex L should be in the rotated triangle. t List the coordinates of the vertices of the rotated triangle. 1), L( ) Compare the coordinates of the rotated triangle with the coordinates of the original triangle. !IIEFI..£Cr '<";- " ;,,--,,,~;;;YY,.,;·,,",· <e_._~,,,· 2. When you reflect or rotate t:.JKL, is the reflected or rotated triangle congruent to the original triangle? How do you know? Describing a Translation A computer artist translates 6.PQR to produce 6.STU. Describe this translation. Determine how many units left or right 6.PQR was translated. t:.STUis _ _ _ units to the right I left of t:.PQR. Determine how many units up or down t:.PQR was translated. t:.STU is _ _ _ units up I down from t:.PQR. So, the artist translated t:.PQR units to the right I left and units up I down. s u 3. Compare the coordinates of the vertices of t:.STU with the coordinates of the vertices of t:.PQR. Chapter 8 333 Lesson 5 PRACTICE Trace t:,FGH on paper and cut it out. Use the cutout triangle for each problem. 1. Translate t:,FGH 5 units to the left. What are the coordinates of the translated triangle? 2. Translate t:,FGH 1 unit to the right and 4 units down. What are the coordinates of the translated triangle? 3. Reflect t:,FGH across the x-axis. The arrow from vertexH shows where vertexH should be in the reflected triangle. What are the coordinates of the reflected triangle? ~2, ), G( 4. Rotate t:,FGH 1800 (or ~ of a full tum) about the origin. The arrow from vertex F shows where vertex F should be in the rotated triangle. What are the coordinates of the rotated triangle? ), G( 5. Compare the coordinates of the vertices of the rotated triangle with the coordinates of the vertices of the original triangle. Use the triangles on the coordinate plane for each problem. 6 6. How is t:,ABC translated to produce t:,JKL? <0 :I: 0 c: g K ::::I s: §jj :::i' 7. How is t:,JKL translated to produce t:,XYZ? x :I: Q,I 8 c: ::l. "tI c: !:!: 8. Error Analysis Blake says that t:,XYZ could be translated 4 units to the right and 6 units up to produce t:,ABC. What error did he make? §: 5° <0 n 0 3 -0 '"::::I '< 9. Reasoning t:,DEFhas sides that measure 3 inches, 4 inches, and 5 inches. t:,GHJhas sides that measure 6 inches, 8 inches, and 10 inches. Could t::.GHlbe a translation of t::.DEF? Explain. Chapter 8 334 Lesson 5 Name _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _ _; Describe each translation. 1. 2. Graph each transformation. 3. Translate MBC 7 units to the right and 1 unit down. 4. Reflect MBC 7 across the x-axis. Chapter 8 335 Practice and Problem Solving 1. Triangle OEF has vertices at 0(0, 0), E(-2, -3), and F(-5, -3). Rotate t,.OEF 90° clockwise about the vertex O. Graph both triangles and write the coordinates of the vertices of the image. Choose the letter for the best answer. 2. What transformation of triangle 1 created triangle 2? A translation 3 units right and 1 unit down B translation 8 units right and 1 unit down C rotation of 180° about the origin D reflection across the y-axis 4. If you reflect triangle 1 across the x-axis, what will be the coordinates of the new triangles? 3. If you rotate triangle 2 90° clockwise about vertex 0, what will be the coordinates of the new triangle? F O'{3, 1), E'(7, 1), F(3, -3) A A'(5, 2), 8'(5, 6), C'(1, 2) G 0'(3, 1), E'(3, -3), F(7, 1) B A'(-5, 0), 8'(-5, -4), C'{-1, 0) H 0'(3, 1), E'{-4, 1), F(-3, 3) C A'(5, -2), 8'{5, -6), C'(1, -2) J 0'(3, 1), E'(-3,3), F(-7, 1) D A'(-5, -2), 8'(-5, -6), C'(-1, -2) Chapter 8 336 Practice and Problem Solving Name _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ _ _.,:;, Problem Solving Connections What's the Pattern? Archeologists are exploring the ruins of an ancient palace. One of the walls contains a tile pattern that has been badly damaged. The archeologists find a scrap of paper with information about one of the tiles. They want to use the information to reconstruct the pattern. What will the complete pattern look like? D Find Angle Measures The figure shows the scrap of paper that the archeologists found. The triangle at the center of the paper, .6.ABC, represents one of the tiles used to make the pattern. A· Name all of the lines that are shown in the figure. B Name three different rays that are shown in the figure. ,C What type of angle pair are LBAC and LDAC? Why? D Let x represent the measure of LBAC. Show how to write and solve an equation to find mLBAC. ,<" E What type of angle is LACG ? How do you know? 337 Let y represent the measure of LACB . Show how to write and solve an equation to find mLACB. What is mLABC? How do you know? D Draw the Triangle The archeologists use what they have discovered to help them draw a full-size version of one of the tiles. lit, The archeologists find that the length of side AC in the tiles was 2 inches. Use a ruler and protractor to draw a triangle that has a 39 angle, a 53 angle, and an included side, AC, that is 2 inches long. Label the vertices of the triangle A, B, and C. 0 0 .... ~ ..'Q. Is there a different triangle that can be drawn with the given conditions? Make another drawing of the triangle using a ruler and protractor. Make the drawing on a piece of cardboard or heavy paper. Label the vertices inside the triangle, as shown at right. Then cut out the triangle. You will use this triangle as a template to reproduce the tile pattern. 338 A Problem· Solving· COll l1 ections D; Begin the Pattern v n;u,' The archeologists reproduce the tile pattern on a piece of t-inch graph paper. Use the coordinate plane on the next page to help you recreate the pattern A .. Place your template triangle on the coordinate plane so that pOint A is at the origin and side AC lies on the positive x-axis. Plot points on the coordinate plane at A, B, and C. What are the coordinates of these points? ...; B· Use the points you plotted to draw .6.ABC on the coordinate plane. 0 C Now rotate .6.ABC 180 counterclockwise about the origin. Draw the rotated triangle on the coordinate plane. What are the coordinates of the vertices of the rotated triangle? D· How do the coordinates of the vertices of the rotated triangle compare with the coordinates of the vertices of the original triangle? Are the two triangles congruent? Why or why not? One of the archeologists suggests that rotating .6.ABC 180 clockwise about the origin might produce a different design. Do you agree or disagree? Why? 0 339 D Answer the Question The archaeologists use additional transformations to complete the pattern. Continue your work on the coordinate plane below. A Translate .6ABC 8 units to the left. Draw the translated triangle on the coordinate plane. What are the coordinates of the vertices of the translated triangle? B Now translate the rotated triangle 8 units to the right. Draw the translated triangle on the coordinate plane. C Now translate D.ABC 5 units left and 4 units down. Draw the translated triangle on the coordinate plane. ·D Continue in this way, filling in and extending the pattern by rotating and translating D.ABC. Draw as much of the pattern as possible on the coordinate plane. 340 Problem SolVIng Connections Name _ _ _ _ _ _ _ _ _ _ Class _______ Date ___,,:i!f Performance Task .,1. Philippe drew this figure in his notebook. a. Name all pairs of vertical angles in the figure. b. Find the value of x. Show your work. . . 2. The diagram shows the intersections of 3 roads. Main Street and Newton Avenue are perpendicular. If a car is traveling north on Newton Avenue, it must turn 140 to take a right onto Fay Drive. At what angle must a car turn to take a right onto Main Street from Fay Drive? Does it matter whether the car is coming from north or south of Main Street? Explain. 0 341 Main 5t Fay Dr Newton Ave '* .,3. Terry chalks the design at right onto the asphalt for a game he's making up. a. The measure of each interior angle of the pentagon is 108°. What are the measures of the angles of the triangle that are adjacent to the pentagon? Explain. h. Find the measure of the third angle of the triangle. Show your work. Cassandra is making a design for a logo. One part of the design is a triangle with two congruent sides. She must draw the triangle with at least one side with length 6 centimeters, and at least one side with length 4 centimeters. a. Sketch two possible figures that Cassandra could use. Label the side lengths in both figures. h. Suppose the side length 4 centimeters is changed to 2 centimeters. How many triangles are possible? Explain using sketches. .'.~ ,..•.... 342 Perforroance Task CHAPTER 8 (~~C~~~N .. ..... ~ ~." ASSESSMENT READINESS Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ _ Date _ __ \ SELECTED RESPONSE 5. The sum of which two angle measures equals the measure of LWLK? Use the figure for problems 1-5. A. LSLYand LnR B. LSLWand LYLR C. LRLK and LSLY D. LRLKand LnR 6. If two angles are supplementary, what is the sum of their measures? 1. Which pair of angles are adjacent angles? A. LSLWand LRLK B. LSLWand LWLK 7. Angle D is a vertical angle to LF. The measure of LD is 53°. What is the measure of LF? C. LSLYand LWLK D. LnRandLnK 2. Which pair of adjacent angles are supplementary angles? F. LRLK and LYLR 8. Which describes the transformation from the original to the image? G. LSLYand LnR H. LRLKand LWLK I~+.!. J. LSLWand LWLR ! y\ i : , j I 3. Which pair of angles are complementary angles? i ( A. LnS and LRLK / B. LYLR and LYLS :j~ji~- C. LSLWand LRLK ;8 D. LWLK and LRLK Ir.. ./ IL. ...-- I ! \: IA IA I V , 'x i\ \ \ '"-.l.\ 8': F. reflection across the x-axis 4. The measure of LRLK is 38°. What is the measure of LSLY? G. translation H. reflection across the y-axis J. rotation Chapter 8 343 Assessment Readiness 11. Name two ways to describe angles TSU and TSR. Explain. 9. Which describes the transformation from the original to the image? A. reflection across the x-axis B. translation 12. Draw a triangle with angle measures of C. rotation 32°, and 45°, and an included side with a length of 2 inches. D. reflection across the y-axis CONSTRUCTED RESPONSE Use the figure for problems 10 and 11. 10. Write and solve an equation to find the measure of LTSU. X" R 5 @ :J: 0 U c: \C .'1 ~ 0 ::J s: 3i 5· !, :J: III ;:: c: 0 ::+ -co c: g ::;. 5· \C 1"\ 0 3 -c III ::J '< Chapter 8 344 Assessment Readiness