24.5 Properties and Conditions for Kites and Trapezoids

Transcription

1 P T S R Locker LSSON 4.5 Properties and onditions for Kites and Trapezoids ommon ore Math Standards The student is expected to: G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices MP.6 Precision Language Objective xplain to a partner how to describe the properties of kites and trapezoids. NGG ssential uestion: What are the properties of kites and trapezoids? The diagonals of a kite are perpendicular; the diagonals of an isosceles trapezoid are congruent; a kite has exactly one pair of congruent opposite angles; an isosceles trapezoid has two pairs of congruent base angles. PRVIW: LSSON PRFORMN TSK View the ngage section online. iscuss the photo. sk students to describe the spider web and in particular its geometrical properties. Then preview the Lesson Performance Task. HROUN S PG 1005 GINS HR Houghton Mifflin Harcourt Publishing ompany Image redits: Larry Mulvehill/orbis Name lass ate 4.5 Properties and onditions for Kites and Trapezoids ssential uestion: What are the properties of kites and trapezoids? xplore xploring Properties of Kites kite is a quadrilateral with two distinct pairs of congruent consecutive sides. In the figure, P PS, and R SR, but R P. Measure the angles made by the sides and diagonals of a kite, noticing any relationships. P Use a protractor to measure PT and TR in the figure. What do your results tell you about the kite s diagonals, PR and S? m PT = 90, m TR = 90 ; the diagonals PR and S are perpendicular. Use a protractor to measure PR and PSR in the figure. How are these opposite angles related? PR PSR Measure PS and RS in the figure. What do you notice? PS RS, so only one pair of opposite angles in the kite are congruent. Use a compass to construct your own kite figure on a separate sheet of paper. egin by choosing a point. Then use your compass to choose points and so that =. T S R Now change the compass length and draw arcs from both points and. Label the intersection of the arcs as point. F Resource Locker Finally, draw the sides and diagonals of the kite. Mark the intersection of the diagonals as point. Module Lesson 5 Name lass ate 4.5 Properties and onditions for Kites and Trapezoids ssential uestion: What are the properties of kites and trapezoids? G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Houghton Mifflin Harcourt Publishing ompany Image redits: Larry Mulvehill/orbis xplore xploring Properties of Kites kite is a quadrilateral with two distinct pairs of congruent consecutive sides. In the figure, P PS, and R SR, but R P. Measure the angles made by the sides and diagonals of a kite, noticing any relationships. Use a protractor to measure PT and TR in the figure. What do your results tell you about the kite s diagonals, PR and S? m PT = 90, m TR = 90 ; the diagonals PR and S are perpendicular. Use a protractor to measure PR and PSR in the figure. How are these opposite angles related? PR PSR PS RS, so only one pair of opposite angles in the kite are congruent. Measure PS and RS in the figure. What do you notice? Use a compass to construct your own kite figure on a separate sheet of paper. egin by choosing a point. Then use your compass to choose points and so that =. Now change the compass length and draw arcs from both points and. Label the intersection of the arcs as point. Resource Finally, draw the sides and diagonals of the kite. Mark the intersection of the diagonals as point. Module Lesson 5 HROVR PGS Turn to these pages to find this lesson in the hardcover student edition. 141 Lesson 4.5

2 G Measure the angles of the kite you constructed in Steps F and the measure of the angles formed by the diagonals. re your results the same as for the kite PRS you used in Steps? Yes, for kite, m = 90 and m = 90, so, again, the diagonals of the kite are perpendicular. lso, for the constructed kite,, but, so, again, one pair of opposite angles are congruent but the other pair are not. Reflect 1. In the kite you constructed in Steps F, look at and. What do you notice? Is this true for and as well? How can you state this in terms of diagonal and the pair of non-congruent opposite angles and? ; yes, ; so, diagonal bisects the pair of non- congruent opposite angles and.. In the kite you constructed in Steps F, look at and. What do you notice? Is this true for and as well? Which diagonal is a perpendicular bisector? ; no, ; so, diagonal is the perpendicular bisector of diagonal, but not vice versa. HROUN S PG 1006 GINS HR XPLOR xploring Properties of Kites INTGRT THNOLOGY Students have the option of doing the kite activity either in the book or online. USTIONING STRTGIS What kind of triangles do the diagonals of a kite form? re any of these triangles congruent? xplain. Right triangles; yes; there are two pairs of congruent triangles by the HL Triangle ongruence Theorem. xplain 1 Using Relationships in Kites The results of the xplore can be stated as theorems. Four Kite Theorems If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral is a kite, then one of the diagonals bisects the pair of non-congruent angles. If a quadrilateral is a kite, then exactly one diagonal bisects the other. P You can use the properties of kites to find unknown angle measures. T S R Module 4 14 Lesson 5 PROFSSIONL VLOPMNT Math ackground kite is a quadrilateral with two distinct pairs of congruent consecutive sides. trapezoid is a quadrilateral with at least one pair of parallel sides. These definitions may not be the same as definitions in other textbooks. Such decisions about definitions are somewhat arbitrary. Variations of definitions do not change the facts of mathematics, but they do change the way the facts are expressed. The decision to use an inclusive definition for trapezoid means that all parallelograms share the properties of trapezoids. The decision to use an exclusive definition for kite means that other quadrilaterals do not necessarily share the properties of kites. Houghton Mifflin Harcourt Publishing ompany XPLIN 1 Using Relationships in Kites INTGRT MTHMTIL PRTIS Focus on Math ommunication MP.3 Point out to students that in the previous lesson they were introduced to the properties of rectangles, rhombuses, and squares. xplain that in this lesson, they will be given a quadrilateral and will learn what conditions can be used to classify it as a kite or a trapezoid. You may want to call on students to read each theorem aloud. Then ask them to explain the theorem in their own words. hallenge students to come up with unique ways to explain the theorems. INTGRT THNOLOGY Have students use geometry software to draw the figures in some of the examples. This will allow them to check their answers. Properties and onditions for Kites and Trapezoids 14

3 USTIONING STRTGIS How do you use the properties of a kite to find the measure of its angles? Since the diagonals of a kite are perpendicular, they form right angles. That makes the acute angles of the right triangles complementary. XPLIN Proving that ase ngles of Isosceles Trapezoids re ongruent HROUN S PG 1007 GINS HR xample 1 In kite, m = 3 and m = 6. Find each measure. m Use angle relationships in. Use the property that the diagonals of a kite are perpendicular, so m = 90. is a right triangle. Therefore, its acute angles are complementary. m + m = 90 Substitute 6 for m, then solve for m. 6 + m = 90 m = 8 Reflect m is also a right triangle. Therefore, its acute angles are complementary. m + m = 90 Substitute 3 for m, then solve for m. m + 3 = 90 m = 58 INTGRT MTHMTIL PRTIS Focus on ommunication MP.3 You may want to review the Parallel Postulate and the orresponding ngles Theorem before you present a plan for the proof. The Parallel Postulate guarantees that there is a unique line through one vertex of the trapezoid that is parallel to one leg of the trapezoid. The segment determined by this line creates a parallelogram, which in turn creates an isosceles triangle (with congruent base angles). The orresponding ngles Theorem applies to the figure because the lines are parallel, and the transitive property will give congruent base angles. Houghton Mifflin Harcourt Publishing ompany 3. From Part and Part, what strategy could you use to determine m? One pair of opposite angles in is congruent, so m = m. lso, m is the sum of m and m. Your Turn 4. etermine m in kite., since exactly one pair of opposite angles are congruent. m = m = m + m = = 86 xplain Proving that ase ngles of Isosceles Trapezoids re ongruent trapezoid is a quadrilateral with at least one pair of parallel sides. The pair of parallel sides of the trapezoid (or either pair of parallel sides if the trapezoid is a parallelogram) are called the bases of the trapezoid. The other two sides are called the legs of the trapezoid. trapezoid has two pairs of base angles: each pair consists of the two angles adjacent to one of the bases. n isosceles trapezoid is one in which the legs are congruent but not parallel. Trapezoid base Isosceles trapezoid leg leg leg base base base Module Lesson 5 leg OLLORTIV LRNING Peer-to-Peer ctivity sk students to work with a partner to make a physical model of a kite and of a trapezoid with paper strips and brads. Have one student make a conjecture about how to find the other three angle measures for the kite, and ask the other student to make a conjecture about how to find the other three angle measures for the trapezoid. sk them to confirm each other s conjecture by measuring the angles. 143 Lesson 4.5

4 Three Isosceles Trapezoid Theorems If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. trapezoid is isosceles if and only if its diagonals are congruent. You can use auxiliary segments to prove these theorems. xample omplete the flow proof of the first Isosceles Trapezoid Theorem. Given: is an isosceles trapezoid with ǁ,. Prove: HROUN S PG 1008 GINS HR USTIONING STRTGIS How is the isosceles triangle in this proof used to show that the base angles of the isosceles trapezoid are congruent? Sample answer: The congruent sides of the isosceles triangle are used to show that its base angles are congruent. Then the proof establishes that the base angles of the isosceles trapezoid are congruent by the transitive property. Given raw intersecting at. Parallel Postulate is a parallelogram. efinition of parallelogram orresponding ngles Theorem Opposite sides of a parallelogram are congruent. Given Substitution Isosceles Triangle Theorem Transitive Property of ongruence Reflect 5. xplain how the auxiliary segment was useful in the proof. Introducing the auxiliary segment breaks the trapezoid into familiar figures, a parallelogram and an isosceles triangle. Then the properties of the simpler figures could be used to prove the theorem about the more complex figure. Houghton Mifflin Harcourt Publishing ompany Module Lesson 5 IFFRNTIT INSTRUTION Multiple Representations Have students make a table of the properties of kites and trapezoids. Have them list the properties of each in their own words and draw a diagram to represent each. Then have students compare different types of quadrilaterals. For example, ask: How are kites and squares alike? Sample answer: oth are quadrilaterals, and both have perpendicular diagonals. Properties and onditions for Kites and Trapezoids 144

5 XPLIN 3 Using Theorems about Isosceles Trapezoids INTGRT MTHMTIL PRTIS Focus on ritical Thinking MP.3 Some math textbooks define a trapezoid as a quadrilateral with exactly one pair of parallel sides. Remind students that this definition is not used here. Parallelograms are a subset of trapezoids because a trapezoid has at least one pair of parallel sides, as they are defined here. Students need to consider this as they are using the theorems about isosceles trapezoids to find segment lengths for trapezoids. HROUN S PG 1009 GINS HR 6. The flow proof in xample only shows that one pair of base angles is congruent. Write a plan for proof for using parallel lines to show that the other pair of base angles ( and ) are also congruent. Plan for Proof: In the isosceles trapezoid, the bases are parallel, so is supplementary to and is supplementary to by the Same Side Interior ngles Postulate. From xample,. So because angles supplementary to angles are Your Turn 7. omplete the proof of the second Isosceles Trapezoid Theorem: If a trapezoid has one pair of base angles congruent, then the trapezoid is isosceles. Given: is a trapezoid with ǁ,. Prove: is an isosceles trapezoid. It is given that ǁ. y the, Parallel Postulate can be drawn parallel to so that intersects at. y the orresponding ngles Theorem,. It is given that, so by substitution,. y the onverse of the Isosceles Triangle Theorem,. opposite sides y definition, is a parallelogram. In a parallelogram, are congruent, so. y the Transitive Property. of ongruence,. Therefore, by definition, isosceles trapezoid is an. USTIONING STRTGIS If you are trying to find the length of one part of a diagonal of an isosceles trapezoid, what information do you need? You need to know that the diagonals of an isosceles trapezoid are congruent. an the bases of a trapezoid be congruent? xplain. Yes, if the trapezoid is also a parallelogram. Houghton Mifflin Harcourt Publishing ompany Image redits: Mariusz Niedzwiedzki/Shutterstock xplain 3 Using Theorems about Isosceles Trapezoids You can use properties of isosceles trapezoids to find unknown values. xample 3 Find each measure or value. railroad bridge has side sections that show isosceles trapezoids. The figure represents one of these sections. = 13. m and = 8.4 m. Find. Use the property that the diagonals are congruent. Use the definition of congruent segments. Substitute 13. for. = 13. = Use the Segment ddition Postulate. + = Substitute 8.4 for and 13. for = 13. Subtract 8.4 from both sides. = 4.8 Module Lesson 5 LNGUG SUPPORT onnect Vocabulary Have students draw a kite and an isosceles trapezoid on poster board and label the diagrams accordingly. For kites, include that their diagonals are perpendicular and that they have one pair of congruent opposite angles. For isosceles trapezoids, include that each pair of base angles is congruent and that the diagonals are congruent. 145 Lesson 4.5

6 Find the value of x so that trapezoid FGH is isosceles. F (x + 1) G HROUN S PG 1010 GINS HR H (3x - 4) For FGH to be isosceles, each pair of base angles are congruent. In particular, the pair at and F are congruent. F Use the definition of congruent angles. m = m F. Substitute 3 x - 4 for m and x + 1 for m F. 3 x - 4 = x + 1 Substract x from both sides and add 4 to both sides. x = 5 Take the square root of both sides. x = 5 or x = -5 Your Turn 8. In isosceles trapezoid PRS, use the Same-Side Interior ngles Postulate to find m R. 9. JL = 3y + 6 and KM = y. etermine the value of y so that trapezoid JKLM is isosceles. J K R M L P 77 Since PRS is isosceles, each pair of base angles must be congruent. P ; m P = m ; 77 = m Using the Same-Side Interior ngles Postulate, m + m R = 180 ; 77 + m R = 180 m R = 103 xplain 4 The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. S For JKLM to be isosceles, its diagonals must be congruent. JL KM JL = KM 3y + 6 = - y 4y = 16 y = 4 Using the Trapezoid Midsegment Theorem midsegment Houghton Mifflin Harcourt Publishing ompany HROUN S PG 1011 GINS HR Module Lesson 5 Properties and onditions for Kites and Trapezoids 146

7 XPLIN 4 Using the Trapezoid Midsegment Theorem Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. XY, XY XY = 1 ( + ) X Y USTIONING STRTGIS In the formula for the length of the midsegment, how do you know which segment lengths to substitute where? Sample answer: The midsegment length will be the length inside the trapezoid connecting the midpoints of the legs; the base lengths will be the lengths of the parallel sides. VOI OMMON RRORS Students may have trouble keeping track of given information, especially when algebraic expressions are involved. ncourage students to color code measures on each diagram. This may help students identify the information needed for their calculations. HROUN S PG 101 GINS HR Houghton Mifflin Harcourt Publishing ompany You can use the Trapezoid Midsegment Theorem to find the length of the midsegment or a base of a trapezoid. xample 4 Find each length. In trapezoid FGH, find XY. Use the second part of the Trapezoid Midsegment Theorem. XY = 1 (H + FG) Substitute 1.5 for H and 10.3 for FG. = 1 ( ) Simplify. = 11.4 In trapezoid JKLM, find JM. P J Use the second part of the Trapezoid Midsegment Theorem. P = 1 ( KL + JM) Substitute 9.8 for P and 8.3 for KL. 9.8 = 1 ( JM) Multiply both sides by = JM Subtract 8.3 from both sides = JM Your Turn 10. In trapezoid PRS, P = RS. Find XY. P = RS 16.8 P X Y S R 16.8 = RS 8.4 = RS XY = 1 (P + RS) = 1 ( ) = 1.6 Module Lesson H K X Y 8.3 L 9.8 F 10.3 G M 147 Lesson 4.5

8 laborate 11. Use the information in the graphic organizer to complete the Venn diagram. Parallelogram Two pairs of parallel sides uadrilateral Kite Two distinct pairs of congruent sides Rectangle Four right angles Square Four right angles and four congruent sides Rectangle Four right angles uadrilateral Parallelogram Two pairs of parallel sides Square Four right angles and four congruent sides Trapezoid t least one pair of parallel sides Rhombus Four congruent sides Trapezoid t least one pair of parallel sides Kite Two distinct pairs of congruent sides Rhombus Four congruent sides What can you conclude about all parallelograms? Possible answer: ll parallelograms are quadrilaterals and all parallelograms are also trapezoids. 1. iscussion The Isosceles Trapezoid Theorem about congruent diagonals is in the form of a biconditional statement. Is it possible to state the two isosceles trapezoid theorems about base angles as a biconditional statement? xplain. HROUN S PG 1013 GINS HR No; the hypotheses and conclusions are not reverses. One has isosceles trapezoid as a hypothesis and each pair of base angles are as a conclusion; the other has a trapezoid and one pair of base angles as a hypothesis and isosceles trapezoid as a conclusion. Houghton Mifflin Harcourt Publishing ompany LORT USTIONING STRTGIS Why can t theorems about the base angles of an isosceles trapezoid be written as biconditionals while theorems about the diagonals can? Sample answer: One starts with an isosceles trapezoid and results in finding two pairs of base angles congruent; the other starts with a trapezoid and one pair of congruent base angles and reasons the trapezoid is isosceles. So, the hypothesis and conclusion of the two theorems are not the reverse of each other. SUMMRIZ TH LSSON Have students list the properties of kites and trapezoids and the difference between a trapezoid and an isosceles trapezoid. Sample answer: The diagonals of a kite are perpendicular; the diagonals of an isosceles trapezoid are congruent; a kite has exactly one pair of congruent opposite angles; an isosceles trapezoid has two pairs of congruent base angles. n isosceles trapezoid has legs that are congruent but not parallel. 13. ssential uestion heck-in o kites and trapezoids have properties that are related to their diagonals? xplain. Yes; the diagonals of a kite are perpendicular, while the diagonals of an isosceles trapezoid are congruent. Module Lesson 5 Properties and onditions for Kites and Trapezoids 148

9 VLUT valuate: Homework and Practice In kite, m = 8 and m = 57. Find each measure. SSIGNMNT GUI oncepts and Skills xplore xploring Properties of Kites xample 1 Using Relationships in Kites xample Proving that ase ngles of Isosceles Trapezoids re ongruent xample 3 Using Theorems about Isosceles Trapezoids xample 4 Using the Trapezoid Midsegment Theorem Practice xercises xercises 1 4 xercises 5 6 xercises 7 10 xercises INTGRT MTHMTIL PRTIS Focus on ommunication MP.3 For some exercises, some students may not realize how to begin solving the problem. Point out that they must first determine if the figure is a kite, a trapezoid, or an isosceles trapezoid. Then suggest that they redraw each figure, marking known properties for that type of quadrilateral. Houghton Mifflin Harcourt Publishing ompany 1. m. m m + m = 90 m + 8 = 90 m = 6 3. m 4. m m = m + m = = 95 Using the first and second Isosceles Trapezoid Theorems, complete the proofs of each part of the third Isosceles Trapezoid Theorem: trapezoid is isosceles if and only if its diagonals are congruent. 5. Prove part 1: If a trapezoid is isosceles, then its diagonals are congruent. Given: is an isosceles trapezoid with,. Prove: m + m = m = 90 m = 33 m = m m = 95 It is given that. y the first Trapezoid Theorem,, and by the Reflexive Property of ongruence,. y the SS Triangle ongruence Theorem,, and by PT,. F Module Lesson Lesson 4.5 xercise epth of Knowledge (.O.K.) Mathematical Practices Recall of information MP.4 Modeling 5 6 Skills/oncepts MP.3 Logic 7 19 Skills/oncepts MP.4 Modeling 0 1 Skills/oncepts MP.4 Modeling 4 Skills/oncepts MP. Reasoning 5 3 Strategic Thinking MP.3 Logic 6 3 Strategic Thinking MP.3 Logic

10 6. Prove part : If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. Given: is a trapezoid with and diagonals. Prove: is an isosceles trapezoid. Statements Reasons 1. raw and F. 1. There is only one line through a given point perpendicular to a given line, so each auxiliary line can be drawn.. F. Two lines perpendicular to the same line are parallel Given 4. F is a parallelogram. 4. efinition of parallelogram (Steps, 3) 5. F 5. If a quadrilateral is a parallelogram, then its opposite sides are congruent Given 7. and F are right angles. 7. efinition of perpendicular lines F HROUN S PG 1014 GINS HR INTGRT MTHMTIL PRTIS Focus on Patterns MP.8 ncourage students to develop their own work patterns when they analyze the many types of quadrilaterals in the exercises, especially when algebraic expressions are involved. For example, they may want to color code the measures on each diagram to help them identify the information needed for their calculations. 8. F 8. HL Triangle ongruence Theorem (Steps 5 7) 9. F 9. PT 10., F 10. lternate Interior ngles Theorem Transitive Property of ongruence (Steps 9, 10) Given Reflexive Property of ongruence SS Triangle ongruence Theorem (Steps 1, 13) PT ngle ddition Postulate 17. is isosceles. 17. If a trapezoid has one pair of base angles congruent, then the trapezoid is isosceles. Houghton Mifflin Harcourt Publishing ompany Module Lesson 5 Properties and onditions for Kites and Trapezoids 150

11 HROUN S PG 1015 GINS HR Use the isosceles trapezoid to find each measure or value. 7. LJ = 19.3 and KN = 8.1. etermine MN. 8. Find the positive value of x so that trapezoid PRS is isosceles. J K N KN + MN = KM; MN = 19.3; MN = In isosceles trapezoid FGH, use the Same-Side Interior ngles Postulate to determine m. 137 H G G H; m G = m H; 137 = m H m + m H = 180 ; m = 180 ; m = 43 Find the unknown segment lengths in each trapezoid. L M L J KM ; L J = KM; 19.3 = KM; F (x + 5) P R S (x + 30) 10. = 3y + 1 and = 7 - y. etermine the value of y so that trapezoid is isosceles. P m = m P x + 5 = x + 30 x = 5 x = 5 = 3y + 1 = 7 - y 5y = 15 y = 3 Houghton Mifflin Harcourt Publishing ompany 11. In trapezoid, find XY. 1. In trapezoid FGH, find FG P P = 1 (H + FG) F X = (.4 + FG) Y =.4 + FG H G XY = ( + ) 1 = ( ) = = FG 13. In trapezoid PRS, P = 4RS. etermine XY. 14. In trapezoid JKLM, P = JK. etermine LM P X P = 4RS; = 4RS; 4.6 = RS XY = (P + RS) 1 = ( ) = 11.5 Y S R M J 7.4 K P L P = JK 7.4 = JK 3.7 = JK 1 P = (JK + LM) = (3.7 + LM) 14.8 = LM 11.1 = LM Module Lesson Lesson 4.5

12 15. etermine whether each of the following describes a kite or a trapezoid. Select the correct answer for each lettered part.. Has two distinct pairs of congruent kite trapezoid consecutive sides. Has diagonals that are perpendicular kite trapezoid. Has at least one pair of parallel sides kite trapezoid. Has exactly one pair of opposite angles kite trapezoid that are congruent. Has two pairs of base angles kite trapezoid 16. Multi-Step omplete the proof of each of the four Kite Theorems. The proof of each of the four theorems relies on the same initial reasoning, so they are presented here in a single two-column proof. Given: is a kite, with and. Prove: (i) ; (ii) ; (iii) bisects and ; (iv) bisects. Statements Reasons 1., 1. Given.. Reflexive Property of ongruence PT 3. SSS Triangle ongruence Theorem (Steps 1, ) HROUN S PG 1016 GINS HR VOI OMMON RRORS Some students may have trouble understanding how a trapezoid is isosceles if and only if its diagonals are congruent. sk them to break up this biconditional statement into two statements to prove. They should see that if the diagonals are congruent, they can use triangle criteria to show congruent triangles and then use corresponding parts of congruent figures to show the trapezoid is isosceles. onversely, they can start with an isosceles trapezoid to show triangles are congruent and then use corresponding parts of congruent figures to show the diagonals are congruent Reflexive Property of ongruence SS Triangle ongruence Theorem (Steps 1, 4, 5) PT PT (Step 3) 10. and 10. PT (Step 3) 8. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 11. bisects and. 11. efinition of angle bisector PT (Step 6) Houghton Mifflin Harcourt Publishing ompany 13. bisects. 13. efinition of segment bisector Module 4 15 Lesson 5 Properties and onditions for Kites and Trapezoids 15

13 17. Given: JKLN is a parallelogram. JKMN is an isosceles trapezoid. Prove: KLM is an isosceles triangle. J K 1. JKLN is a parallelogram. (Given);. KL N NJ M (Opposite sides of a parallelogram are congruent.); 3. JKMN is an isosceles trapezoid. (Given); 4. NJ MK (efinition of isosceles trapezoid); 5. KL KM (Transitive Property of ongruence); 6. KLM is an isosceles triangle. (efinition of isosceles triangle) L lgebra Find the length of the midsegment of each trapezoid x 6x 1 4x = (1 + 6x) 8x = 1 + 6x x = 6, so 4x = 4 (6) = 4 3y - 7 y + 6 y + 3 y + 6 = 1 ( (y + 3) + (3y - 7) ) (y + 6) = (y + 3) + (3y - 7) y + 1 = 4y = y, so y + 6 = = 14 HROUN S PG 1017 GINS HR 0. Represent Real-World Problems set of shelves fits an attic room with one sloping wall. The left edges of the shelves line up vertically, and the right edges line up along the sloping wall. The shortest shelf is 3 in. long, and the longest is 40 in. long. Given that the three shelves are equally spaced vertically, what total length of shelving is needed? The shelves form 1 a trapezoid and the middle shelf forms its midsegment. middle shelf: (3 in in.) = 36 in. Total: = 108 in. Houghton Mifflin Harcourt Publishing ompany 1. Represent Real-World Problems common early stage in making an origami model is known as the kite. The figure shows a paper model at this stage unfolded. The folds create four geometric kites. lso, the 16 right triangles adjacent to the corners of the paper are all congruent, as are the 8 right triangles adjacent to the center of the paper. Find the measures of all four angles of the kite labeled (the point is the center point of the diagram). Use the facts that and that the interior angle sum of a quadrilateral is 360. The 8 congruent central triangles are isosceles right triangles; so, m = (45) = 90. The four angles 1 at each corner of the paper are congruent, so m = m = (90 ) =.5 and m = = Since, m = m = x ; then: 90 + x x = 360, x = 360, and x = So m = m = Module Lesson Lesson 4.5

14 . nalyze Relationships The window frame is a regular octagon. It is made from eight pieces of wood shaped like congruent isosceles trapezoids. What are m, m, m, and m in trapezoid? xtend 1 and to their intersection. Since m = (360 ) = 45, and since, 8 m + m + m = 180 m + 45 = 180 m = 135 m = 67.5 m = 67.5 y the Same-Side Int. s Post., m + m = 180, so m = = Since, m = xplain the rror In kite, m = 66 and m = 59. Terrence is trying to find m. He knows that bisects, and that therefore. He reasons that, so that m = (59 ) = 118, and that because they are opposite angles in the kite, so that m = 118. xplain Terrence s error and describe how to find m. Terrence mistakenly reasoned that ; only one pair of opposite angles in a kite are congruent, and they are adjacent to the bisected diagonal, not the bisecting diagonal. To find m : Since and are complementary, m can be found by = 4. Then, since so that, m is twice m, or (4 ) = omplete the table to classify all quadrilateral types by the rotational symmetries and line symmetries they must have. Identify any patterns that you see and explain what these patterns indicate. uadrilateral ngle of Rotational Symmetry none Number of Line Symmetries kite 1 non-isosceles trapezoid none 0 isosceles trapezoid parallelogram none rectangle rhombus square Houghton Mifflin Harcourt Publishing ompany The quadrilaterals with rotational symmetry are parallelograms and special cases of parallelograms; the more restricted cases of quadrilaterals tend to have more line symmetries, up to the square with 4; there are two pairs of quadrilateral types with the same symmetries, kites and isosceles trapezoids, and rectangles and rhombuses. Module Lesson 5 Properties and onditions for Kites and Trapezoids 154

15 JOURNL Have students draw an isosceles trapezoid. Have them label the angle measures in terms of x and write a justification for each measure. HROUN S PG 1018 GINS HR H.O.T. Focus on Higher Order Thinking 5. ommunicate Mathematical Ideas escribe the properties that rhombuses and kites have in common, and the properties that are different. Rhombuses and kites both have pairs of congruent consecutive sides, but in a rhombus, this is because all four sides are congruent. In a kite, two distinct pairs of consecutive sides are congruent. The diagonals of both types of quadrilaterals are perpendicular. In a kite, exactly one diagonal is bisected by the other, while in a rhombus, each diagonal bisects the other. Finally, in a kite, exactly one pair of opposite angles are congruent, while both pairs of opposite angles of a rhombus are congruent. 6. nalyze Relationships In kite, triangles and can be rotated and translated, identifying with and joining the remaining pair of vertices, as shown in the figure. Why is this process guaranteed to produce an isosceles trapezoid? Suggest a process guaranteed to produce a kite from an isosceles trapezoid, using figures to illustrate your process. ' Houghton Mifflin Harcourt Publishing ompany = ' = ' Sample answer for the case of producing an isosceles trapezoid from a kite where the congruent angles are obtuse: y the definition of a kite and the Reflexive Property of ongruence,,,, and. Rotate and translate the triangles and so that sides and coincide, to produce the quadrilateral shown. raw and 'F perpendicular to the line containing at points and F. and 'F are parallel because they are perpendicular to the same line. ''' by SSS, so ''' because corresponding parts of congruent figures are congruent. Then ''F because supplements of congruent angles = = F are congruent. 'F' because they are both right angles. lso, '' (from the kite), so ''F by S. So, 'F because corresponding parts of congruent figures are congruent. uadrilateral 'F is a parallelogram, because one pair of opposite sides are parallel and congruent. y the definition of a parallelogram, ' and F (which includes (or '' )) are parallel, so ' is a trapezoid because it has a pair of parallel sides. onsider the lines and '' cut by transversal F. Then and ''' are corresponding angles. ecause is obtuse, its supplement,, is acute. ut ''' is obtuse, which means that and ''' cannot be congruent. Thus, is not parallel to '' and ' is not a parallelogram. trapezoid that is not a parallelogram but has congruent legs is isosceles, so ' is an isosceles trapezoid. Sample answer: JKLM is an isosceles trapezoid, with JK ML but JK KL or JM K L. Rotate and translate triangles JKL and MLK so that sides JL and M'K' coincide. This can be done because the diagonals of an isosceles trapezoid are congruent. J M The resulting figure has two distinct pairs of congruent sides, JK M'L' (given) and KL L'K' K (Reflexive Property of ongruence), so the figure is a kite. J = M' Module Lesson 5 L = K' L' 155 Lesson 4.5

16 Lesson Performance Task This model of a spider web is made using only isosceles triangles and isosceles trapezoids. INTGRT MTHMTIL PRTIS Focus on Math onnections MP.1 In the model of the spider web, = 6 cm and = 3 cm. Without referring to the circumference of either the outer or the inner circle, explain how you know that the radius of the inner circle is half the radius of the outer circle. y the converse of the Triangle Midsegment Theorem, is the midpoint of. So,, a radius of the inner circle, has half the measure of, a radius of the outer circle. a. ll of the figures surrounding the center of the web are congruent to figure. Find m. xplain how you found your answer. b. Find m and m. c. Find m and m. d. Find m and m. a..5 ; 16 triangles congruent to surround the center of the web, making up a total of =.5 b. m =.5 and is an isosceles triangle. So, and m = m 180 = m + m = m + m = m = m Thus, each angle measures c. ecause the angles form a linear pair, then m + m = 180. So m = 180 and m = In an isosceles trapezoid, base angles are congruent, so m = d. In isosceles trapezoid,. Then corresponding angles are congruent and and. So m = m = Houghton Mifflin Harcourt Publishing ompany INTGRT MTHMTIL PRTIS Focus on Math onnections MP.1 xplain how you know that is parallel to. Sample answer: m = = m, so. So, ǁ because corresponding angles and are congruent. Module Lesson 5 XTNSION TIVITY The Lesson Performance Task deals with triangles and trapezoids in spider webs. Have students research other examples that illustrate quadrilaterals in the animal world, the plant world, or both. For each example students find, they should make a sketch, identify the quadrilateral, and provide additional information they feel is relevant, particularly as it relates to geometry. Scoring Rubric Points: Student correctly solves the problem and explains his/her reasoning. 1 Point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 Points: Student does not demonstrate understanding of the problem. Properties and onditions for Kites and Trapezoids 156