6.3 Showing Quadrilaterals are

Transcription

1 6.3 howing Quadrilaterals are arallelograms Goal how that a quadrilateral is a parallelogram. Key Words parallelogram p. 30 Geo-ctivity aking arallelograms ut two straws to form two congruent pairs. 2 artly unbend two paper clips, link their smaller ends, and insert the larger ends into two cut straws, as shown. Join the rest of the straws to form a quadrilateral with opposite sides congruent, as shown. 3 hange the angles of your quadrilateral. Is your quadrilateral always a parallelogram? The Geo-ctivity above describes one way to show that a quadrilateral is a parallelogram. tudent Help THEOE 6.6 and 6.7 TUY TI The theorems in this lesson are the converses of the theorems in Lesson 6.2. Theorem 6.6 Words If both pairs of opposite sides of a quadrilateral are congruent, then the ymbols If Q&* c &* and Q&* c &*, then Q is a parallelogram. Theorem 6.7 Words If both pairs of opposite angles of a quadrilateral are congruent, then the ymbols If a ca and aq ca, then Q is a parallelogram. 36 hapter 6 Quadrilaterals

2 Itudent Help I LZONE.O OE EXLE ore examples at classzone.com EXLE Use Opposite ides Tell whether the quadrilateral is a olution The quadrilateral is not a parallelogram. It has two pairs of congruent sides, but opposite sides are not congruent. EXLE 2 Use Opposite ngles Tell whether the quadrilateral is a olution The quadrilateral is a parallelogram because both pairs of opposite angles are congruent. Use Opposite ides and Opposite ngles In Exercises and 2, tell whether the Explain your reasoning L 5 K J 3. In quadrilateral WXYZ, WX 5, YZ 20, XY 5, and ZW 20. Is WXYZ a parallelogram? Explain your reasoning. THEOE 6.8 Words If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the y x x x y 80 ymbols If ma maq 80 and maq ma 80, then Q is a parallelogram. 6.3 howing Quadrilaterals are arallelograms 37

3 EXLE 3 Use onsecutive ngles Tell whether the Explain your reasoning. a. U V b. E F c T 85 W H G olution a. au is supplementary to at and av ( ). o, by Theorem 6.8, TUVW is a parallelogram. b. ag is supplementary to af ( ), but ag is not supplementary to ah ( ). o, EFGH is not a parallelogram. c. a is supplementary to a ( ), but you are not given any information about a or a. Therefore, you cannot conclude that is a parallelogram. THEOE 6.9 Words If the diagonals of a quadrilateral bisect each other, then the ymbols If Q&* c &*** and &** c &**, then Q is a parallelogram. EXLE 4 Use iagonals Tell whether the Explain your reasoning. a. b. J K L olution 2 a. The diagonals of JKL bisect each other. o, by Theorem 6.9, JKL is a parallelogram. b. The diagonals of Q do not bisect each other. o, Q is not a parallelogram hapter 6 Quadrilaterals

4 Use onsecutive ngles and iagonals Tell whether the Explain your reasoning. 4. G F H 30 E K L N U V 5 5 X T 4 4 T W You have learned five ways to show that a quadrilateral is a parallelogram. UY HOWING QUILTEL I LLELOG efinition of parallelogram, p. 30 how that both pairs of opposite sides are parallel. Theorem 6.6, p. 36 how that both pairs of opposite sides are congruent. Theorem 6.7, p. 36 how that both pairs of opposite angles are congruent. Theorem 6.8, p. 37 how that one angle is supplementary to both of its consecutive angles. y x x x y 80 Theorem 6.9, p. 38 how that the diagonals bisect each other. 6.3 howing Quadrilaterals are arallelograms 39

5 6.3 Exercises Guided ractice Vocabulary heck In Exercises 3, name all the sides or angles of gefgh that match the description.. Opposite sides are parallel. E F 2. Opposite angles are congruent. 3. onsecutive angles are supplementary. H G kill heck 4. Explain why every parallelogram is a quadrilateral, but not every ecide whether you are given enough information to show that the Explain your reasoning ractice and pplications Extra ractice ee p Using Opposite ides Tell whether the quadrilateral is a Using Opposite ngles Tell whether the quadrilateral is a Homework H lp Example : Exs. 8 0 Example 2: Exs. 3 Example 3: Exs. 4 6 Example 4: Exs. 7 9 Using onsecutive ngles Tell whether the quadrilateral is a hapter 6 Quadrilaterals

6 Using iagonals Tell whether the Explain your reasoning. icycles icycle Gears When you change gears on a bicycle, the derailleur moves the chain to the new gear. For the derailleur at the right,.8 cm, 3.6 cm,.8 cm, and 3.6 cm. Explain why &* and &** are always parallel when the derailleur moves Error nalysis What is wrong with the student s argument below? EILLEU (named from the French word meaning to derail ) move the chain to change gears. pplication Links LZONE.O quadrilateral that has one pair of sides congruent and the other pair of sides parallel is always a parallelogram. You be the Judge 22. Three of the interior angles of a quadrilateral have measures of 75, 75, and 05. Is this enough information to conclude that the quadrilateral is a parallelogram? Explain your answer. 23. Visualize It! Explain why the following method of drawing a parallelogram works. tate a theorem to support your answer. Use a ruler to draw 2 raw another 3 a segment and its segment so the midpoint. midpoints coincide. onnect the endpoints of the segments. 24. hallenge If one pair of opposite sides of a quadrilateral is both congruent and parallel, is the quadrilateral a parallelogram? Explain your reasoning. 6.3 howing Quadrilaterals are arallelograms 32

7 tudent Help KILL EVIEW To review the formula for finding slope, see p EXLE oordinate Geometry Use the slopes of the segments in the diagram to determine if the y (2, 4) (6, 4) olution (, 0) 4 (5, 0) x Lines and line segments are parallel if they have the same slope. lope of &*: lope of 2 &*: lope of &*: lope of &*: NWE The slopes of &* and &* are the same, so &* &*. The slopes of &*and &* are the same, so &* &*. oth pairs of opposite sides are parallel, so is a parallelogram. oordinate Geometry Use the slopes of the segments in the diagram to determine if the 25. y 26. G(, 4) H(5, 4) y Œ(2, 5) (5, 5) F(2, ) J(6, ) x (0, 2) (4, 2) x tandardized Test ractice 27. ulti-tep roblem uppose you shoot a pool ball as shown below and it rolls back to where it started. The ball bounces off each wall at the same angle at which it hits the wall. a. The ball hits the first wall at an angle of 63. o maef maeh 63. Explain why mafe 27. b. Explain why mafg 63. c. What is magh? maeh? d. Find the measure of each interior angle of EFGH. What kind of quadrilateral is EFGH? How do you know? 322 hapter 6 Quadrilaterals

8 ixed eview Finding ngle easures Find the measure of a. (Lesson 6.) Finding easures Find the measure in gjkl. (Lesson 6.2) 3. Find mak. 32. Find maj. L Find L. 34. Find KL. lgebra kills Evaluating Expressions Evaluate the expression for the given value of the variable. (kills eview, p. 670) 35. 2x 7 when x y 3 when y 2 K 4 J m when m b when b a when a c 5 when c x x 2 when x q2 2 when q n 3 4n when n p p 2 when p 3 Quiz ecide whether the figure is a polygon. If so, tell what type. If not, explain why. (Lesson 6.) Find the values of the variables in the parallelogram. (Lesson 6.2) 4. y 5. x y 6. y 2 6 x x 9 22 z 5 Tell whether the Explain your reasoning. (Lesson 6.3) howing Quadrilaterals are arallelograms 323

9 Technology ctivity 6.3 aking arallelograms Question How can you use the angle measures in a quadrilateral to show that it is a parallelogram? Explore raw quadrilateral. 2 easure the angles of the quadrilateral. 3 rag the vertices until ma ma and ma ma. ma 74 ma 87 ma 9 ma 80 ma 60 ma 20 ma 60 ma 20 tudent Help TUY TI ecall that parallel lines and parallel segments have the same slope. Think bout It. Find the slopes of &*, &*, &*, and &*. What do you notice about the slopes of opposite sides? 2. What do the slopes tell you about the sides of the quadrilateral? 3. What kind of figure is quadrilateral? Use your results from Exercise 2 to help you. 4. What theorem does this exploration illustrate? 5. Extension raw quadrilateral EFGH. raw segments EG&* and FH&*. onstruct a point I at the intersection of EG&* and FH&*. easure EI&*, IG&*, FI&*, and IH&*. rag any of the vertices of EFGH so that EI IG and FI IH. What do you notice? What theorem does this illustrate? 324 hapter 6 Quadrilaterals