6 5 Trapezoids And Kites-Books Pdf

6 5 Trapezoids and Kites
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Quick Check 1 In the isosceles trapezoid m S 70 P Q 2 Teach. Find m P m Q and m R , 110 110 70, 70 , S R Guided Instruction. 2 EXAMPLE Real World Connection 1 EXAMPLE Error Prevention. Some students may think the base, Architecture The second ring of the ceiling shown at the left is made from. angles of an isosceles trapezoid, congruent isosceles trapezoids that create the illusion of circles What are the. have vertices only on the bottom , measures of the base angles of these trapezoids . side This misconception stems from, Each trapezoid is part of an isosceles triangle whose base angles are the acute base the common use of the word base.
angles of the trapezoid The isosceles triangle has a vertex angle that is half as large to mean the side of a figure on. as one of the 20 angles at the center of the ceiling which it rests Point out that each. isosceles trapezoid has two bases , which may lie in any orientation . and two pairs of base angles , 1, 2 EXAMPLE Careers. Architects design modern office, The measure of each angle at the center of the ceiling is 360. 20 or 18 , buildings using not only, trapezoids and squares but also. The measure of 1 is 18, 2 or 9 triangles circles and ellipses .
You are looking up at The measure of each acute base angle is 18022 9 or 85 5 . Harbour Centre Tower in, Connection to Engineering. Vancouver Canada The measure of each obtuse base angle is 180 85 5 or 94 5 . Have students find how a, Quick Check 2 A glass ceiling like the one above has 18 angles meeting at the center instead of 20 keystone is used and how its. What are the measures of the base angles of the trapezoids in its second ring 85 95 shape relates to this lesson . Visual Learners, Like the diagonals of parallelograms the diagonals of an isosceles trapezoid have a When proving that the diagonals. special property of an isosceles trapezoid are, congruent have students. separately draw and label the, Key Concepts Theorem 6 16.
overlapping triangles ABC and, The diagonals of an isosceles trapezoid are congruent DCB to help them see how the. parts correspond and why the, triangles are congruent . Proof Proof of Theorem 6 16 PowerPoint, Given Isosceles trapezoid ABCD with AB DC A D Additional Examples. Prove AC DB, 1 XYZW is an isosceles trapezoid , It is given that AB DC Because the base B C and m X 156 Find m Y m Z . angles of an isosceles trapezoid are congruent and m W . ABC DCB By the Re exive Property of Congruence BC BC Then X Y. by the SAS Postulate ABC DCB Therefore AC DB by CPCTC 156 . W Z, mlY 156 , Another special quadrilateral that is not a parallelogram is a kite The diagonals of mlZ mlW 24.
a kite like the diagonals of a rhombus are perpendicular A proof of this for a kite. next page is quite like its proof for a rhombus at the top of page 330 2 Half of a spider s web is. shown below formed by layers, Lesson 6 5 Trapezoids and Kites 337 of congruent isosceles trapezoids . Find the measures of the angles, in ABDC , A, Advanced Learners L4 English Language Learners ELL B. After students read the proof of Theorem 6 16 have Draw several trapezoids in different orientations and. C, them write a paragraph explaining whether the have students identify the bases legs and base angles D. diagonals of an isosceles trapezoid bisect each other Emphasize that the parallel sides are called bases and. are independent of their orientation , learning style verbal learning style verbal mlA mlB 75 337. mlC mlD 105, Math Tip, After students read the proof Key Concepts Theorem 6 17.
of Theorem 6 17 ask Are both, diagonals bisected No the figure The diagonals of a kite are perpendicular . then would be a parallelogram , 3 EXAMPLE Teaching Tip. Proof Proof of Theorem 6 17 T,Discuss as a class how to prove. Given Kite RSTW with TS TWand RS RW, DBA DBC using SSS . Prove TR SW, S W, PowerPoint Both T and R are equidistant from S and W By the Converse Z.
of the Perpendicular Bisector Theorem T and R lie on the. Additional Examples perpendicular bisector of SW Since there is exactly one line. 3 Find m 1 m 2 and m 3 through any two points Postulate 1 1 TR must be the. perpendicular bisector of SW Therefore TR SW R,in the kite . S, 1, You can use Theorem 6 17 to nd angle measures in kites . D, R T, 3 2, 72 3 EXAMPLE Finding Angle Measures in Kites. B, U, ml1 72 ml2 90 Find m 1 m 2 and m 3 in the kite 32 . 3, ml3 18 m 1 90 Diagonals of a kite are perpendicular .
90 m 2 32 180 Triangle Angle Sum Theorem 1 2, A C. Resources, 122 m 2 180 Simplify , Daily Notetaking Guide 6 5 L3. Daily Notetaking Guide 6 5 m 2 58 Subtract 122 from each side . Adapted Instruction L1 ABD CBD by SSS , D, By CPCTC m 3 m DBC 32 . Closure Quick Check 3 Find m 1 m 2 and m 3 in the kite . 90 46 44,Draw and label an isosceles 46 1, 2,trapezoid a convex kite and. their diagonals Then write 3,congruence statements for all.
pairs of triangles that you can, prove congruent Students EXERCISES For more exercises see Extra Skill Word Problem and Proof Practice . should find three pairs of, congruent triangles for each Practice and Problem Solving. figure , A Practice by Example Each trapezoid is isosceles Find the measure of each angle . Example 1 1 2 3 , 3 2 1 69 69 2 3, page 336 111 111. GO for, Help, 77 1 3, 2, 1 49 , 77 103 103 49 131 131.
4 Z 5 Q R 6 B C, Y 105 , 75 75, 65 60 , P S A D, 105 . X 115 115 65 120 120 60, W, 338 Chapter 6 Quadrilaterals. 17 Answers may vary Sample ,338, Example 2, page 337 . 7 Design Each patio umbrella is made, of eight panels that are congruent 3 Practice. isosceles triangles with parallel, stripes A sample panel is shown Assignment Guide.
at the right The vertex angle, of the panel measures 42 1 A B 1 39. 7a isosc trapezoids a Classify the quadrilaterals. C Challenge 40 44, shown as blue stripes on, the panel . Test Prep 45 50, b Find the measures of, Mixed Review 51 56. the quadrilaterals , interior angles , 69 69 111 111 Homework Quick Check. To check students understanding, Example 3 Find the measures of the numbered angles in each kite .
108 108 of key skills and concepts go over, page 338 . 8 22 9 45 10 1 Exercises 6 14 29 37 38 , 1 90 68 3 90 45 . 54 90 , 2 1 2 45 Diversity, 2, Exercise 7 The word umbrella. 11 12 90 40 13 comes from a Latin word mean , 2, 1 2 90 ing shaded area or shadow . 64 , 13 90 55 90 55 35 3 50 1 3 35 1 suggesting protection against the.
5 3 rain or sun Ask whether students,14 90 52 38 37 53. know the word for umbrella in, 15 90 90 90 90 46 34 2 4 other languages For example . 90 26 90, 56 44 56 44 the Spanish word paraguas,16 112 112 14 15 16 literally means for water . 2 4 9 10 2 and a sombrilla is a parasol , 38 1 53 34 1 2 46 1. 3 5 6 34 5 Exercise 10 Discuss ways to prove, 468 m 1 m 2 .
7 8, Exercise 38 Have students work, B Apply Your Skills 17 Open Ended Sketch two kites that are not congruent but with the diagonals of together to write this proof . one congruent to the diagonals of the other See margin Even with the Plan the proof. is complex and worthy of class, 18 The perimeter of a kite is 66 cm The length of one of its sides is 3 cm less than discussion . twice the length of another Find the length of each side of the kite . 12 12 21 21, 19 Critical Thinking If KLMN is an isosceles trapezoid is it possible for KM to. bisect LMN and LKN Explain See margin , x 2 Algebra Find the value of the variable in each isosceles trapezoid . 20 T W 21 M N 22 B C GPS Guided Problem Solving L3. 12 15 15 60 3x 15 Enrichment L4, 45 3x Reteaching L2.
60 5x L O, S R Adapted Practice L1, A D Practice Name Class Date L3. 23 T U 24 R S 25 Q R Practice 6 5 Trapezoids and Kites. GO nline 3 4 1, Find the measures of the numbered angles in each isosceles trapezoid . 1 2 2 , 99 , 3 2, Homework Help, 121 , 62 1, U T 1 1. 2, Visit PHSchool com SU x 1, 4 , 2, 5 2 6 2, V P S. S, 1 67 , Web Code aue 0605, TR 2x 3 QS x 5, 96 1 1.
79 , TV 2x 1, G O 3 CS 060, RP 3x 3, Algebra Find the value s of the variable s in each isosceles trapezoid . US x 2, 7 3x 3 8 6x 20 9 L M, y , 7x 2y 5, x 1 4x . x 5 O N, Find the measures of the numbered angles in each kite . Lesson 6 5 Trapezoids and Kites 339 10 , 1, 101 , 2. 11 12 , 44 , 1, 80 , 2 2, 1 65 , Pearson Education Inc All rights reserved .
48 , 19 Explanations may vary CPCTC opp L O N or a square This 13 . 1, 3, 2, 27 , 14 , 1, 59 , 2, 15 , 1, 51 , 3, Sample If both are L and N are opp but contradicts what is gien 87 2. bisected then this KLMN is isos both so KM cannot bisect. Algebra Find the value s of the variable s in each kite . 16 17 5x 1 18 y 9 4x 13 , combined with KM O pairs of base s are LMN and LKN 10x 6 . 2x , 3x , 8x , 5x 15 , y , KM by the Reflexive also O By the Trans . Prop means kKLM O Prop all 4 angles are O , kKNM by SAS By so KLMN must be a rect 339.
4 Assess Reteach x 2 Algebra Find the value s of the variable s in each kite . 26 27 3x 5 28 , PowerPoint, y y 6x , Lesson Quiz 2y 20 . x 6 , 28 4x 30 3x ,Use isosceles trapezoid ABCD for 2x 2. 2x 4 ,Exercises 1 and 2 x 35 y 30 x 18 y 108, A B. Bridge Design A quadrilateral is formed by the beams of the bridge at the left . Isosc trapezoid all the, D C 29 Classify the quadrilateral Explain your reasoning large rt appear to be O . 1 If m A 45 find 30 Find the measures of the other interior angles of the quadrilateral . m B m C and m D mlB 112 68 68, 45 Critical Thinking Can two angles of a kite be as follows Explain .
mlC mlD 135 31 33 See below left , 31 opposite and acute 32 consecutive and obtuse. 2 If AC 3x 16 and, 33 opposite and supplementary 34 consecutive and supplementary. BD 10x 86 find x 10, Use kite GHIJ for Exercises 3 6 35 opposite and complementary 36 consecutive and complementary. 112 , H 34 36 See margin , Exercises 29 30 37 Writing A kite is sometimes de ned as a quadrilateral with two pairs of. 4, consecutive sides congruent Compare this to the de nition you learned in.
50 31 Yes the O can be, G I obtuse Lesson 6 1 Are parallelograms trapezoids rhombuses rectangles or squares. 3 1, special kinds of kites according to the changed de nition Explain See margin . 32 Yes the O can be, obtuse as well as 38 Developing Proof The plan suggests a proof of Theorem 6 15 Write a proof. one other l GPS that follows the plan See back of book . 9 , 2 33 Yes if 2 O are rt Given Isosceles trapezoid ABCD with AB DC A D. they are suppl The, other 2 are also Prove B C and BAD D 1.
B C, E, J suppl Plan Begin by drawing AE 6 DC to form. 4, 43 D is any point on BN parallelogram AECD so that AE DC AB . 3 Find m 1 90 B C because B 1 and 1 C Also BAD D, such that ND u BN and. 4 Find m 2 9 D is below N because they are supplements of the congruent angles B and C . 5 Find m 3 81, Proof Write a proof Use the given gure with additional lines as needed . 6 Find m 4 40, 39 Given Isosceles trapezoid TRAP with TR PA.
Prove RTA APR See margin T P, Alternative Assessment C Challenge 40 Given Isosceles trapezoid TRAP with TR PA . BI is the perpendicular bisector of RA intersecting. 41 It is one half the sum, Have students work in pairs to RA at B and TP at I See margin . of the lengths of the R A, write answers to the following bases draw a diag Prove BI is the perpendicular bisector of TP . questions of the trap to form, How are a kite and a rhombus 2 The bases B For a trapezoid consider the segment joining the midpoints of the two given. similar How are they and b of the trap are segments How are its length and the lengths of the two parallel sides of the. different each a base of a n trapezoid related Justify your answer . Then the segment, How are an isosceles trapezoid, joining the midpts 41 the two nonparallel sides 42 the diagonals.
and a rectangle similar How of the non n sides is See left B. are they different the sum of the See margin , 43 BN is the perpendicular bisector of AC at N . midsegments of the, Describe the set of points D for which ABCD is a kite . This sum is See above left A N C, 1 1 1 Proof 44 Prove that the angles formed by the noncongruent sides. 2 B 2 b 2 B b , 34 No if two consecutive of a kite are congruent Hint Draw a diagonal of the kite . are suppl then another See back of book , pair must be also 340 Chapter 6 Quadrilaterals.
because one pair of opp , is O Therefore the, opp would be O 35 Yes the O must be 39 Answers may vary 3 TR O PA Given . which means the figure 45 or 135 each Sample Draw TA and RP . would be a and not a 36 No if two consecutive 4 RA O RA Refl Prop . were compl then the 1 isosc trapezoid TRAP of O , kite Given . kite would be concave , 37 Rhombuses and squares 5 kTRA O kPAR SSS . 2 TA O PR Diagonals, would be kites since opp 6 lRTA O lAPR. 340 sides can be O also of an isosc trap are O CPCTC . Test Prep, Test Prep, Resources, Multiple Choice 45 Which statement is true for every trapezoid B For additional practice with a.
A Exactly two sides are congruent B Exactly two sides are parallel variety of test item formats . C Opposite angles are supplementary D The diagonals bisect each other Standardized Test Prep p 361. 46 Which statement is true for every kite J Test Taking Strategies p 356. F Opposite sides are congruent G At least two sides are parallel Test Taking Strategies with. H Opposite angles are supplementary J The diagonals are perpendicular Transparencies. 47 Two consecutive angles of a trapezoid are right angles Three of the. following statements about the trapezoid could be true Which statement. CANNOT be true A, A The two right angles are base angles . B The diagonals are not congruent , C Two of the sides are congruent . D No two sides are congruent , 48 Quadrilateral EFGH is a kite What is the G. value of x H, F 15 75 , F H, G 70, x 20 , H 85, E. J 160, Short Response 49 In the trapezoid at the right BE 5 2x 2 8 .
A D, DE 5 x 2 4 and AC 5 x 1 2 , 49 2 a 2x 8 x 4 a Write and solve an equation for x E. B C 42 It is one half the, x 2 OR b Find the length of each diagonal . equivalent difference of the lengths, equation x 7 50 Diagonal RB of kite RHBW forms an equilateral triangle with two of the of the bases from. sides m BWR 40 Draw and label a diagram showing the diagonal and. b 9 9 Ex 41 the length of the, the measures of all the angles Which angles of the kite are largest . segment joining the, 1 one computational See margin .
error midpts of the non n, sides is 12 B b By the. Side Splitter Thm the, Mixed Review middle part of this. segment joins the, Lesson 6 4 Find the indicated angle measures for the rhombus 126 3 midpts of the diags . GO for, Help 51 m 1 126 52 m 2 27 53 m 3 27 Each outer segment. 1, measures 21 b So the, 2, length of the segment.
Lesson 5 2 x 2 Algebra Find the values indicated connecting the midpts . F A of the diags is, 54 a a 55 a x 1, b FG 2a 3 b CD D B 2 B b . c GH E G c BC 50 2 , a 1 2x 24 7x 9, a 4 a 3 R, b 5 H b 30 C. c 5 c 30 60 70 , Lesson 4 2 56 State the postulate that justi es the B C D H 60 40 W. statement ABC DEF SAS, 60 70 , A F E B, lHRW and lHBW. 1 incorrect diagram, lesson quiz PHSchool com Web Code aua 0605 Lesson 6 5 Trapezoids and Kites 341 OR no work shown.
40 Draw BI as described 4 kTRB O kPAB SAS 9 kTBI O kPBI SAS . then draw BT and BP 5 BT O BP CPCTC 10 lBIT O lBIP CPCTC . 6 lRBT O lABP 11 lBIT and lBIP are, 1 TR O PA Given CPCTC rt O supp are. 2 lR O lA Base of 7 lTBI O kPBI Compl rt , isosc trap are O of O are O 12 TI O PI CPCTC . 3 RB O AB Def of 8 BI O BI Refl Prop 13 BI is bis of TP .

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