9.5 Properties and Conditions for Kites and Trapezoids

Transcription

1 Name lass ate 9.5 Properties and onditions for Kites and Trapezoids ssential uestion: What are the properties of kites and trapezoids? Resource Locker 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. P T R Measure the angles made by the sides and diagonals of a kite, noticing any relationships. S 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? Houghton Mifflin Harcourt Publishing ompany Image redits: Larry Mulvehill/orbis Use a protractor to measure PR and PSR in the figure. How are these opposite angles related? 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. F Finally, draw the sides and diagonals of the kite. Mark the intersection of the diagonals as point. Module Lesson 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? 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? 2. 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? 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 T R Houghton Mifflin Harcourt Publishing ompany S You can use the properties of kites to find unknown angle measures. Module Lesson 5

3 xample 1 In kite, m = 32 and m = 62. 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 62 for m, then solve for m m = 90 m = 28 m is also a right triangle. Therefore, its acute angles are complementary. m + m = Substitute 32 for m, then solve for m. m + = m = Reflect 3. From Part and Part, what strategy could you use to determine m? Your Turn 4. etermine m in kite. Houghton Mifflin Harcourt Publishing ompany xplain 2 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

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 2 omplete the flow proof of the first Isosceles Trapezoid Theorem. Given: _ is an isosceles trapezoid with ǁ, _. Prove: Given raw intersecting at. Parallel Postulate is a efinition of orresponding ngles Theorem Opposite sides of a parallelogram are congruent. Substitution Isosceles Triangle Theorem Property of ongruence Reflect 5. xplain how the auxiliary segment was useful in the proof. Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

5 6. The flow proof in xample 2 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. 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. y the, _ 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, _. y definition, is a parallelogram. In a parallelogram, are congruent, so _. y the Transitive Property. of ongruence, _. Therefore, by definition, is an. 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. = 13.2 m and = 8.4 m. Find. Use the property that the diagonals are congruent. Use the definition of congruent segments. Substitute 13.2 for. = 13.2 = Use the Segment ddition Postulate. + = Substitute 8.4 for and 13.2 for = 13.2 Subtract 8.4 from both sides. = 4.8 Module Lesson 5

6 Find the value of x so that trapezoid FGH is isosceles. F (2x ) G H (3x 2-4) For FGH to be isosceles, each pair of base angles are congruent. In particular, the pair at and are congruent. Use the definition of congruent angles. m = m. Substitute for m and for m. = Substract from both sides and add to both sides. x 2 = Take the square root of both sides. x = or x = Your Turn 8. In isosceles trapezoid PRS, use the Same-Side Interior ngles Postulate to find m R. 9. JL = 3y + 6 and KM = 22 y. etermine the value of y so that trapezoid JKLM is isosceles. J K R M L P 77 S xplain 4 The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. Using the Trapezoid Midsegment Theorem midsegment Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

7 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 ( + ) 2 X Y 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 H X Y F 10.3 G Use the second part of the Trapezoid Midsegment Theorem. XY = 1_ (H + FG) 2 Substitute 12.5 for H and 10.3 for FG. = 1_ ( ) 2 Simplify. = 11.4 In trapezoid JKLM, find JM. K 8.3 L 9.8 M P J Houghton Mifflin Harcourt Publishing ompany Use the second part of the Trapezoid Midsegment Theorem. P = 1_ 2 ( + JM) Substitute for P and for. = 1_ 2 ( + JM) Multiply both sides by 2. = + JM Your Turn Subtract from both sides. = JM 10. In trapezoid PRS, P = 2RS. Find XY Y R P X S Module Lesson 5

8 laborate 11. Use the information in the graphic organizer to complete the Venn diagram. Rectangle Four right angles Rhombus Four congruent sides Parallelogram Two pairs of parallel sides uadrilateral Trapezoid t least one pair of parallel sides Square Four right angles and four congruent sides Kite Two distinct pairs of congruent sides Two pairs of parallel sides Four right angles Four right angles and four congruent sides Four congruent sides Two distinct pairs of congruent sides t least one pair of parallel sides What can you conclude about all parallelograms? 12. 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. Houghton Mifflin Harcourt Publishing ompany 13. ssential uestion heck-in o kites and trapezoids have properties that are related to their diagonals? xplain. Module Lesson 5

9 valuate: Homework and Practice In kite, m = 28 and m = 57. Find each measure. 1. m 2. m 3. m 4. m 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. Houghton Mifflin Harcourt Publishing ompany 5. Prove part 1: If a trapezoid is isosceles, then its diagonals are congruent. Given: _ is an _ isosceles trapezoid with,. Prove: It is given that. y the first Trapezoid Theorem,, and by the Reflexive Property of ongruence,. y the SS Triangle ongruence Theorem,, and by,. F Module Lesson 5

10 6. Prove part 2: 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. F 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. 2. F 2. Two lines perpendicular to the same line are parallel Given 4. F is a parallelogram. 4. (Steps 2, 3) _ If a quadrilateral is a parallelogram, then its opposite sides are congruent efinition of perpendicular lines 8. F 8. HL Triangle ongruence Theorem (Steps 5 7) 9. F , F10. lternate Interior ngles Theorem Transitive Property of ongruence (Steps 9, 10) Given (Steps 12, 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

11 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. K N L (2x 2 + 5) R J M S P (x ) 9. In isosceles trapezoid FGH, use the Same-Side Interior ngles Postulate to determine m. F 10. = 3y + 12 and = 27-2y. etermine the value of y so that trapezoid is isosceles. H G 137 Find the unknown segment lengths in each trapezoid. 11. In trapezoid, find XY. 12. In trapezoid FGH, find FG. Houghton Mifflin Harcourt Publishing ompany 13.3 X P Y F H G 13. In trapezoid PRS, P = 4RS. etermine XY. 14. In trapezoid JKLM, P = 2JK. etermine LM. M Y J R P X S K P L Module Lesson 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 1., 1. Given _ 2. Reasons 2. Reflexive Property of ongruence (Steps 1, 2) PT Reflexive Property of ongruence SS Triangle ongruence Theorem (Steps 1, 4, 5) If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular (Step 3) 10. and 10. (Step 3) _ 11. bisects and.11. efinition of Houghton Mifflin Harcourt Publishing ompany PT (Step 6) 13. bisects.13. Module Lesson 5

13 17. Given: JKLN is a parallelogram. JKMN is an isosceles trapezoid. Prove: KLM is an isosceles triangle. J K N M L lgebra Find the length of the midsegment of each trapezoid x 6x 3y - 7 y + 6 y 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 32 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? Houghton Mifflin Harcourt Publishing ompany 21. 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. Module Lesson 5

14 22. 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? 23. 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 = 2 (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. 24. 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 Number of Line Symmetries kite 1 non-isosceles trapezoid isosceles trapezoid none parallelogram 180 rectangle rhombus square Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

15 H.O.T. Focus on Higher Order Thinking 25. ommunicate Mathematical Ideas escribe the properties that rhombuses and kites have in common, and the properties that are different. 26. 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? ' Next, suggest a process guaranteed to produce a kite from an isosceles trapezoid, using figures to illustrate your process. = ' = ' Houghton Mifflin Harcourt Publishing ompany Module Lesson 5

16 Lesson Performance Task This model of a spider web is made using only isosceles triangles and isosceles trapezoids. 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. Houghton Mifflin Harcourt Publishing ompany Module Lesson 5