24.2 Conditions for Parallelograms

Transcription

1 Locker LESSON 24 2 onditions for Parallelograms Name lass ate 242 onditions for Parallelograms Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram? ommon ore Math Standards The student is expected to: G-O11 Prove theorems about parallelograms lso G-O10, G-SRT5 Mathematical Practices MP2 Reasoning Language Objective Explain to a partner how to identify the opposite sides, opposite angles, and consecutive angles and sides of a quadrilateral Explore Proving the Opposite Sides riterion for a Parallelogram You can prove that a quadrilateral is a parallelogram by using the definition of a parallelogram That is, you can show that both pairs of opposite sides are parallel However, there are other conditions that also guarantee that a quadrilateral is a parallelogram Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram omplete the proof of the theorem Given: and Resource Locker ENGGE Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram? Opposite sides are congruent; opposite sides are parallel; opposite angles are congruent; one angle is supplementary to both its consecutive angles; a pair of opposite sides are congruent and parallel; diagonals bisect one another PREVIEW: LESSON PERFORMNE TSK View the Engage section online iscuss the photo e sure students understand that the stars in a constellation are at various distances from Earth, and that the shape the constellation appears to take is purely a line-of-sight effect Then preview the Lesson Performance Task Houghton Mifflin Harcourt Publishing ompany Prove: is a parallelogram raw diagonal Why is it helpful to draw this diagonal? rawing the auxiliary line, diagonal, creates two triangles to use in the proof Use triangle congruence theorems and corresponding parts to complete the proof that the opposite sides are parallel so the quadrilateral is a parallelogram Statements Reasons 1 raw 1 Through any two points, there is exactly one line 2 2 Reflexive Property of ongruence 3 ; 3 Given 4 4 SSS Triangle ongruence Theorem 5 ; 5 PT 6 ; 6 onverse of the lternate Interior ngles Theorem 7 is a parallelogram 7 efinition of parallelogram Module Lesson 2 Name lass ate 24 2 onditions for Parallelograms Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram? G-O11 Prove theorems about parallelograms lso G-O10, G-SRT5 Explore Proving the Opposite Sides riterion Houghton Mifflin Harcourt Publishing ompany for a Parallelogram You can prove that a quadrilateral is a parallelogram by using the definition of a parallelogram That is, you can show that both pairs of opposite sides are parallel However, there are other conditions that also guarantee that a quadrilateral is a parallelogram Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram omplete the proof of the theorem Given: and Prove: is a parallelogram raw diagonal Why is it helpful to draw this diagonal? Use triangle congruence theorems and corresponding parts to complete the proof that the opposite sides are parallel so the quadrilateral is a parallelogram Statements Reasons Resource rawing the auxiliary line, diagonal, creates two triangles to use in the proof Reflexive Property of ongruence Given SSS Triangle ongruence Theorem PT onverse of the lternate Interior ngles Theorem efinition of parallelogram 1 raw 1 Through any two points, there is exactly one line ; ; 5 6 ; 6 7 is a parallelogram 7 Module Lesson 2 HROVER PGES Turn to these pages to find this lesson in the hardcover student edition 1203 Lesson 24 2

2 It is possible to combine the theorem from the Explore and the definition of a parallelogram to state the following condition for proving a quadrilateral is a parallelogram You will prove this in the exercises Theorem If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram EXPLORE Proving the Opposite Sides riterion for a Parallelogram Reflect 1 iscussion quadrilateral has two sides that are 3 cm long and two sides that are 5 cm long student states that the quadrilateral must be a parallelogram o you agree? Explain isagree; you can only conclude that the quadrilateral is a parallelogram if you know that the congruent sides are opposite each other Explain 1 Proving the Opposite ngles riterion for a Parallelogram You can use relationships between angles to prove that a quadrilateral is a parallelogram Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram INTEGRTE TEHNOLOGY Students have the option of doing the proof activity either in the book or online QUESTIONING STRTEGIES What is a sufficient condition to prove that a quadrilateral is a parallelogram? Either both pairs of opposite sides are parallel, or both pairs of opposite sides are congruent Example 1 Prove that a quadrilateral is a parallelogram if its opposite angles are congruent Given: and Prove: is a parallelogram m + m + m + m = 360 by the Polygon ngle Sum Theorem From the given information, m = m and m = m y substitution, m + m + m + m = 360 or 2m + 2m = 360 ividing both sides by 2 gives m + m = 180 Therefore, and are onverse of the Same-Side Interior ngles Postulate supplementary and so by the similar argument shows that, so is a parallelogram by the definition of parallelogram Reflect 2 What property or theorem justifies dividing both sides of the equation by 2 in the above proof? ivision Property of Equality Module Lesson 2 PROFESSIONL EVELOPMENT Houghton Mifflin Harcourt Publishing ompany EXPLIN 1 Proving the Opposite ngles riterion for a Parallelogram INTEGRTE MTHEMTIS PRTIES Focus on Math onnections MP1 The proof of this criterion depends on students understanding the hypothesis and conclusion of the statement oth pairs of opposite angles are given to be congruent, and that is part of the hypothesis Having these angles congruent leads to both pairs of opposite sides of the quadrilateral being parallel, which satisfies the definition of a parallelogram Stating that the quadrilateral is a parallelogram is the conclusion Math ackground In this lesson, students extend their earlier work with the properties of parallelograms to establish sufficient conditions for concluding that a quadrilateral is a parallelogram quadrilateral is a parallelogram if its opposite sides are parallel, if its opposite sides are congruent, if its opposite angles are congruent, if its diagonals bisect each other, or if one pair of opposite sides is congruent and parallel It is worth emphasizing that the statements to prove in the previous lesson all have the form If a quadrilateral is a parallelogram, then [property] In contrast, the statements to prove in this lesson all have the form, If [condition], then the quadrilateral is a parallelogram QUESTIONING STRTEGIES Suppose only one pair of opposite angles of a quadrilateral are congruent an you still conclude that the quadrilateral is a parallelogram? Explain No; in this case, the quadrilateral could be a kite onditions for Parallelograms 1204

3 EXPLIN 2 Proving the isecting iagonals riterion for a Parallelogram Explain 2 Proving the isecting iagonals riterion for a Parallelogram You can use information about the diagonals in a given figure to show that the figure is a parallelogram Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram INTEGRTE MTHEMTIL PRTIES Focus on Reasoning MP2 Students should read this criterion carefully and pay attention to the sufficient conditions given and to how the criterion is proved The proof depends on students using the SS ongruence riterion, the Vertical ngles Theorem, and the Opposite Sides riterion Pairs of triangles formed by the diagonals are congruent, and this makes the opposite sides of the quadrilateral congruent, so that the Opposite Sides riterion (for a parallelogram) applies QUESTIONING STRTEGIES How does the proof of this criterion use the Opposite Sides riterion? The proof uses the SS triangle congruence criteria and the fact that corresponding parts of congruent triangles are congruent to show that the opposite sides of a quadrilateral are congruent Then the Opposite Sides riterion is applied to state that the quadrilateral must be a parallelogram In the proof of the Opposite Sides riterion, which theorem justifies that both pairs of opposite sides are parallel? Once the alternate interior angles are stated to be congruent, the onverse of the lternate Interior ngles Theorem states that the lines are parallel Houghton Mifflin Harcourt Publishing ompany Example 2 Prove that a quadrilateral whose diagonals bisect each other is a parallelogram Given: E E and E E Reflect Prove: is a parallelogram Statements 1 E E, E E 1 Given Reasons 2 E E, E E 2 Vertical angles are congruent 3 E E, E E 3 SS Triangle ongruence Theorem 4, 4 PT 5 is a parallelogram 5 3 ritique Reasoning student claimed that you can also write the proof using the SSS Triangle ongruence Theorem since and o you agree? Justify your response No; in order to use the fact that opposite sides are congruent, you must first know that the quadrilateral is a parallelogram Explain 3 Using a Parallelogram to Prove the oncurrency of the Medians of a Triangle Sometimes properties of one type of geometric figure can be used to recognize properties of another geometric figure Recall that you explored triangles and found that the medians of a triangle are concurrent at a point that is 2 of the distance from each vertex to the midpoint of the opposite side 3 You can prove this theorem using one of the conditions for a parallelogram from this lesson Example 3 omplete the proof of the oncurrency of Medians of a Triangle Theorem Given: If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram Prove: The medians of are concurrent at a point that is 2 of the distance from 3 each vertex to the midpoint of the opposite side Module Lesson 2 OLLORTIVE LERNING Peer-to-Peer ctivity ivide students into pairs Have one student draw different quadrilaterals (kites, rhombuses, rectangles, etc) Have the other student determine whether the quadrilateral meets any of the criteria for a parallelogram, and list them fter a while, have the students switch roles E 1205 Lesson 24 2

4 Let be a triangle such that M is the midpoint of and N is the midpoint of Label the point where the two medians intersect as P raw MN MN is a midsegment of because it connects the midpoints of two sides of the triangle 1 MN is parallel to and MN = 2 by the Triangle Midsegment Theorem Let Q be the midpoint of P and let R be the midpoint of P raw QR QR is a midsegment of P because it connects the midpoints of two sides of the triangle 1 QR is parallel to and QR = 2 by the Triangle Midsegment Theorem So, you can conclude that MN = QR by substitution and that MN QR because both segments are parallel to the same segment Now draw MQ and NR and consider quadrilateral MQRN Quadrilateral MQRN is a parallelogram because a pair of opposite sides are congruent and parallel Since the diagonals of a parallelogram bisect each other, then QP = NP lso, Q = QP since Q is the midpoint of P Therefore, Q = QP = distance from to N This shows that point P is located on N at a point that is 2 of the 3 y similar reasoning, the diagonals of a parallelogram bisect each other, so RP = MP lso, R = RP since R is the midpoint of P Therefore, R = RP = distance from to M NP MP This shows that point P is located on M at a point that is 2 of the 3 You can repeat the proof using any two medians of The same reasoning shows that the medians from vertices and intersect at a point that is also 2 of the distance from to M, so this point must 3 also be point P This shows that the three medians intersect at a unique point P and that the point is 2 3 of the distance from each vertex to the midpoint of the opposite side M M M P P N Q N P Q N R R Houghton Mifflin Harcourt Publishing ompany EXPLIN 3 Using a Parallelogram to Prove the oncurrency of the Medians of a Triangle QUESTIONING STRTEGIES How is the Triangle Midsegment Theorem used in proving the concurrency of the medians in a triangle? The Triangle Midsegment Theorem is used to show that the opposite sides of a quadrilateral are parallel This creates a parallelogram within the triangle Then, since the diagonals of a parallelogram bisect each other, the parallelogram in turn is used to prove the concurrency of the medians in the triangle VOI OMMON ERRORS Students may be confused when asked to locate the midsegments in a triangle that already has the medians labeled for this proof Have students use one color pencil to mark the medians, a second color pencil to mark the midsegments, and a third color pencil to mark the congruent segments on the diagonals of the parallelogram Module Lesson 2 IFFERENTITE INSTRUTION Kinesthetic Experience Have students use raw spaghetti to demonstrate the theorems in the lesson For example, ask them to try to form a parallelogram with opposite sides that are congruent but not parallel, or vice versa This emphasizes that, in the statement that a quadrilateral with a pair of opposite sides parallel and congruent leads to a parallelogram, the same pair of opposite sides must be congruent and parallel onditions for Parallelograms 1206

5 EXPLIN 4 Verifying Figures re Parallelograms QUESTIONING STRTEGIES How is the Opposite Sides riterion used to show that the given quadrilateral is a parallelogram? lgebra is used to show that the opposite sides are congruent after the given values are substituted into the expression The figure meets the criterion and must be a parallelogram VOI OMMON ERRORS Some students may not know which criterion to use when proving that a quadrilateral is a parallelogram Explain that they should study the given information carefully to determine which condition is best to apply For example, if they are given congruent angles, then they should look only at the two criteria that deal with angles: both pairs of opposite angles are congruent, or one angle is supplementary to both of its consecutive angles Houghton Mifflin Harcourt Publishing ompany Reflect 4 In the proof, how do you know that point P is located on N at a point that is 2 of the 3 distance from to N? Possible answer: Since Q = QP = PN, N is divided into three segments of equal Explain 4 Verifying Figures re Parallelograms You can use information about sides, angles, and diagonals in a given figure to show that the figure is a parallelogram Example 4 Show that each quadrilateral is a parallelogram for the given values of the variables x = 7 and y = 4 Step 1 Find and length (Q = QP = PN) So, points P and Q divide N into thirds Therefore, since P = Q + QP = N + 3 N = N, then point P is 2 of the distance from to N 3 3 = x + 14 = = 21 = 3x = 3 (7) = 21 Step 2 Find and = 5y - 4 = 5 (4) - 4 = 16 = 2y + 8 = 2 (4) + 8 = 16 So, = and = is a parallelogram since both pairs of opposite sides are congruent z = 11 and w = 45 Step 1 Find m F and m H (9z + 19) (9(11) + 19) = 118 m F = = m H = (11z - 3) = (11(11) - 3) = 118 Step 2 Find m E and m G m E = (12w + 8) (12(45) + 8) = 62 = m G = (14w - 1) (14(45) - 1) = 62 = So, m F = m H and m E = m G EFGH is a parallelogram since both pairs of opposite angles are congruent E x y - 4 2y + 8 3x F (9z + 19) (14w - 1) (12w + 8) (11z - 3) H G Reflect 5 What conclusions can you make about FG and EH in Part? Explain FG is parallel and congruent to EH ; EFGH is a parallelogram and opposite sides of a parallelogram are parallel and congruent Module Lesson 2 LNGUGE SUPPORT onnect Vocabulary Encourage students to pay close attention to the key words in each theorem, such as opposite, parallel, and congruent Have them write each of the theorems on note cards, illustrate the theorems, and highlight these key words 1207 Lesson 24 2

6 Your Turn Show that each quadrilateral is a parallelogram for the given values of the variables 6 a = 24 and b = 9 7 x = 6 and y = 35 Q R (10b - 16) (9b + 25) 7a 2a + 12 P S PQ = 7a = 7 (24) = 168 RS = 2a + 12 = 2 (24) + 12 = 168 m Q = (10b - 16) = (10(9) - 16) = 74 m R = (9b + 25) = (9(9) + 25) = 106 So, PQ = RS lso, PQ RS since same-side interior angles are supplementary PQRS is a parallelogram since a pair of opposite sides are parallel and congruent Elaborate J K 2x + 2 6y + 1 N 8y - 6 4x - 10 M L JN = 2x + 2 = 2 (6) + 2 = 14 LN = 4x - 10 = 4 (6) - 10 = 14 KN = 8y - 6 = 8 (35) - 6 = 22 MN = 6y + 1 = 6 (35) + 1 = 22 So JN = LN and KN = MN JKLM is a parallelogram since the diagonals bisect each other 8 How are the theorems in this lesson different from the theorems in the previous lesson, Properties of Parallelograms? In the previous lesson, you begin with a quadrilateral that is known to be a parallelogram and each theorem states a property of the parallelogram In this lesson, you begin with a quadrilateral and each theorem states conditions that guarantee that the quadrilateral is a parallelogram ELORTE QUESTIONING STRTEGIES How do the theorems in this lesson differ from the ones in the previous lesson? They are converses Here we know the properties, and are proving the quadrilateral is a parallelogram; previously, we knew the figure was a parallelogram and we were proving properties SUMMRIZE THE LESSON Two opposite angles in a quadrilateral each measure 103 an you conclude that this quadrilateral is a parallelogram? Explain why or draw a counterexample No, we can t, because only one pair of opposite angles is congruent Possible counterexample: 9 Why is the proof of the oncurrency of the Medians of a Triangle Theorem in this lesson and not in the earlier module when the theorem was first introduced? The proof involves creating a quadrilateral that needs to be shown is a parallelogram This could not be done before this lesson 10 Essential Question heck-in escribe three different ways to show that quadrilateral is a parallelogram Houghton Mifflin Harcourt Publishing ompany Possible answer: Since right angles are congruent, opposite angles of are congruent Therefore, is a parallelogram Opposite sides of are congruent, so is a parallelogram is supplementary to, so Therefore, is a parallelogram since a pair of opposite sides are congruent and parallel Module Lesson 2 onditions for Parallelograms 1208

7 EVLUTE SSIGNMENT GUIE oncepts and Skills Explore Proving the Opposite Sides riterion for a Parallelogram Example 1 Proving the Opposite ngles riterion for a Parallelogram Example 2 Proving the isecting iagonals riterion for a Parallelogram Example 3 Using a Parallelogram to Prove the oncurrency of the Medians of a Triangle Example 4 Verifying Figures are Parallelograms Practice Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercises 5 12 INTEGRTE MTHEMTIL PRTIES Focus on Modeling MP4 Some students may benefit from a hands-on approach for exploring which quadrilaterals are parallelograms Have students copy a larger version of the quadrilateral in an exercise onto a sheet of paper and then cut it out Have them use a ruler to verify that the opposite sides are congruent, or a protractor to verify that the opposite angles are congruent, so that the criteria for the quadrilateral to be a parallelogram are satisfied Houghton Mifflin Harcourt Publishing ompany Evaluate: Homework and Practice 1 You have seen a proof that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Write the proof as a flow proof Given: and Prove: is a parallelogram Possible answer: Given Given,, Through any two points, there is exactly one line Reflexive Property of ongruence SSS Triangle ongruence Theorem PT is a parallelogram efinition of parallelogram raw onverse of the lternate Interior ngles Theorem Online Homework Hints and Help Extra Practice Module Lesson 2 Exercise epth of Knowledge (OK) Mathematical Practices Skills/oncepts MP3 Logic Recall MP5 Using Tools Skills/oncepts MP2 Reasoning 19 3 Strategic Thinking MP3 Logic 20 3 Strategic Thinking MP4 Modeling 21 3 Strategic Thinking MP2 Reasoning 1209 Lesson 24 2

8 2 You have seen a proof that if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Write the proof as a two-column proof Given: and Prove: is a parallelogram Statements 1 and 1 Given Reasons 2 m + m + m + m = Polygon ngle Sum Theorem 3 m + m + m + m = 360 ; m + m + m + m = m + 2m = 360 ; 2m + 2m = m + m = 180 ; m + m = and are supplementary; and are supplementary 7 and 3 Substitution Property of Equality 4 istributive Property 5 ivision Property of Equality 6 efinition of supplementary angles 7 onverse of the Same-Side Interior ngles Postulate 8 is a parallelogram 8 efinition of parallelogram INTEGRTE MTHEMTIL PRTIES Focus on Reasoning MP2 Point out that only one criterion of the six listed below needs to be met to show that a quadrilateral is a parallelogram: oth pairs of opposite sides are parallel One pair of opposite sides is parallel and congruent oth pairs of opposite sides are congruent oth pairs of opposite angles are congruent One angle is supplementary to both of its consecutive angles The diagonals bisect each other 3 You have seen a proof that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram Write the proof as a paragraph proof Given: E E and E E Prove: is a parallelogram Possible answer: It is given that E E and E E Since vertical angles are congruent, E E and E E Therefore, E E and E E by the SS Triangle ongruence Theorem Since corresponding parts of congruent triangles are congruent, and So, is a parallelogram because if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram 4 omplete the following proof of the Triangle Midsegment Theorem Given: is the midpoint of, and E is the midpoint of Prove: E, E = 1 2 Extend E to form F such that E FE Then draw F, as shown E E Houghton Mifflin Harcourt Publishing ompany E F Module Lesson 2 onditions for Parallelograms 1210

9 It is given that E is the midpoint of, so E E y the Vertical ngles Theorem, E EF So, E EF by the SS Triangle ongruence Theorem Since corresponding parts of congruent triangles are congruent, F is the midpoint of, so y the Transitive Property of ongruence, F Houghton Mifflin Harcourt Publishing ompany lso, since corresponding parts of congruent triangles are congruent, E FE So, F by onverse of the lternate Interior ngles Theorem This shows that F is a parallelogram because a pair of opposite sides are congruent and parallel y the definition of parallelogram, E is parallel to Since opposite sides of a parallelogram are congruent, = F E FE, so E = 1 F and by substitution, E = Show that each quadrilateral is a parallelogram for the given values of the variables 5 x = 4 and y = 9 6 u = 8 and v = 35 y + 4 M J 5x + 7 K 2y - 5 6x + 3 L JK = 5x + 7 = 5 (4) + 7 = 27 LM = 6x + 3 = 6 (4) + 3 = 27 KL = 2y 5 = 2 (9) 5 = 13 MJ = y + 4 = = 13 So, JK = LM and KL = MJ JKLM is a parallelogram etermine if each quadrilateral must be a parallelogram Justify your answer 7 8 No One pair of opposite sides are parallel different pair of opposite sides are congruent 6v - 10 E = 2u + 3 = 2(8) + 3 = 19 E = 3u 5 = 3(8) 5 = 19 2u + 3 E = 4v 3 = 4(35) 3 = 11 E = 6v 10 = 6(35) 10 = 11 E 3u - 5 So E = E and E = E is a parallelogram 4v - 3 Yes the third pair of angles in the triangles are congruent So, both pairs of opposite angles are congruent Module Lesson Lesson 24 2

10 9 10 Yes pair of alternate interior angles are congruent, so a pair of opposite sides are parallel The same pair of opposite sides are congruent by SS and PT Yes The 73 angle is supplementary to both of the 107 angles This shows that both pairs of opposite sides are parallel by the onverse of the Same-Side Interior ngles Postulate ommunicate Mathematical Ideas Kalil wants to write the proof that the medians of a triangle are concurrent at a point that is 2 of the distance 3 from each vertex to the midpoint of the opposite side He starts by drawing PQR and two medians, PK and QL He labels the point of intersection as point J, as shown What segment should Kalil draw next? What conclusions can he make about this segment? Explain No One pair of opposite sides are congruent and one diagonal is bisected by the other No You are only given the measures of the four angles formed by the intersecting diagonals of the quadrilateral He should draw KL KL is a midsegment of PQR because it connects the midpoints of two sides of the triangle KL is parallel to PQ and KL = 1 2 PQ by the Triangle Midsegment Theorem P L J R K Q PEER-TO-PEER ISUSSION sk students to discuss with a partner how to create a graphic organizer that will display all the criteria they have learned that make a quadrilateral a parallelogram Then have them create the graphic organizer and make sure they include the following criteria: Opposite sides parallel Opposite sides congruent Opposite angles congruent onsecutive angles supplementary iagonals bisecting each other One pair of sides congruent and parallel 14 ritical Thinking Jasmina said that you can draw a parallelogram using the following steps 1 raw a point P 2 Use a ruler to draw a segment that is 1 inch long with its midpoint at P 3 Use the ruler to draw a segment that is 2 inches long with its midpoint at P 4 Use the ruler to connect the endpoints of the segments to form a parallelogram oes Jasmina s method always work? Is there ever a time when it would not produce a parallelogram? Explain The method always works as long as the second segment does not overlap the first segment The two segments are diagonals Since the diagonals bisect each other, the quadrilateral is a parallelogram Houghton Mifflin Harcourt Publishing ompany Module Lesson 2 onditions for Parallelograms 1212

11 15 ritique Reasoning Matthew said that there is another condition for parallelograms He said that if a quadrilateral has two congruent diagonals, then the quadrilateral is a parallelogram o you agree? If so, explain why If not, give a counterexample to show why the condition does not work No The figure shows a counterexample The diagonals are congruent, but the quadrilateral does not have two pairs of parallel opposite sides, so it is not a parallelogram 16 parallel rule can be used to plot a course on a navigation chart The tool is made of two rulers connected at hinges to two congruent crossbars, and You place the edge of one ruler on your desired course and then move the second ruler over the compass rose on the chart to read the bearing for your course If, why is always parallel to? Possible answer: and are opposite sides of a quadrilateral It is given that and Therefore, is a parallelogram So, is always parallel to 17 Write a two-column proof to prove that a quadrilateral with a pair of opposite sides that are parallel and congruent is a parallelogram Given: and Prove: is a parallelogram (Hint: raw ) Statements Reasons Houghton Mifflin Harcourt Publishing ompany Image redits: drian Muttitt/lamy 1 raw Given 3 3 Given Through any two points, there is exactly one line 4 4 lternate Interior ngles Theorem 5 5 Reflexive Property of ongruence 6 6 SS Triangle ongruence Theorem 7 7 PT 8 is a parallelogram 8 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Module Lesson Lesson 24 2

12 18 oes each set of given information guarantee that quadrilateral JKLM is a parallelogram? Select the correct answer for each lettered part JN = 25 cm, JL = 50 cm, KN = 13 cm, KM = 26 cm Yes No MJL KLJ, JM LK Yes No JM JK, KL LM Yes No MJL MLJ, KJL KLJ Yes No E JKN LMN Yes No a Yes; The diagonals of the quadrilateral bisect each other, so JKLM is a parallelogram b Yes; pair of opposite sides are both parallel and congruent, so JKLM is a parallelogram c No; none of the conditions for a parallelogram are met d No; none of the conditions for a parallelogram are met e Yes; pair of opposite sides are both parallel and congruent, so JKLM is a parallelogram HOT Focus on Higher Order Thinking 19 Explain the Error student wrote the two-column proof below Explain the student s error and explain how to write the proof correctly Given: 1 2, E is the midpoint of 1 E 2 Prove: is a parallelogram J M N K L INTEGRTE MTHEMTIL PRTIES Focus on ritical Thinking MP3 t this point, students may be thinking of the theorems from Lesson 9 1 and Lesson 9 2 as if and only if statements (biconditionals) about the relationship between quadrilaterals and parallelograms Many of the theorems can be combined to be biconditionals, but make sure students can still focus on which phrase is the hypothesis and which phrase is the conclusion when they apply the theorems For the exercises in this lesson, the hypothesis includes a statement that a quadrilateral has certain given attributes The conclusion of the statement is that the quadrilateral is a parallelogram Statements Given Reasons 2 E is the midpoint of 2 Given 3 E E 3 efinition of midpoint 4 E E 4 Vertical angles are congruent 5 E E 5 S Triangle ongruence Theorem 6 6 orresponding parts of congruent triangles are congruent 7 is a parallelogram 7 If a pair of opposite sides of a quadrillateral are congruent, then the quadrillateral is a parallelogram Possible answer: The student used an invalid reason in Step 7 The student should also show that This is true because 1 2 and these are alternate interior angles Then the student can conclude that is a parallelogram because a pair of opposite sides are both parallel and congruent Houghton Mifflin Harcourt Publishing ompany Module Lesson 2 onditions for Parallelograms 1214

13 JOURNL Have students describe how to use the measures of the sides of a quadrilateral to determine whether it is a parallelogram Houghton Mifflin Harcourt Publishing ompany Image redits: eth Reitmeyer/Houghton Mifflin Harcourt 20 Persevere in Problem Solving The plan for a city park shows that the park is a quadrilateral with straight paths along the diagonals For what values of the variables is the park a parallelogram? In this case, what are the lengths of the paths? If the diagonals bisect each other, then RSTU will be a parallelogram, so assume RV = TV and SV = UV 16y 10 = 12y y 10 = 50 4y = 60 (5x + 2y) m y = 15 R (12y + 50) m S V 5x + 2y = 2x + 4y U (16y - 10) m 5x + 2 (15) = 2x + 4 (15) T 5x + 30 = 2x + 60 (2x + 4y) m 3x + 30 = 60 3x = 30 x = 10 UV = 12y + 50 = 12 (15) + 50 = 230 m US = 2UV = 2 (230) = 460 m RV = 5x + 2y = 5 (10) + 2 (15) = 80 m RT = 2RV = 2 (80) = 160 m The lengths of the paths are 230 m and 160 m 21 nalyze Relationships When you connect the midpoints of the consecutive sides of any quadrilateral, the resulting quadrilateral is a parallelogram Use the figure below to explain why this is true (Hint: raw a diagonal of ) M L J Possible answer: When you draw, you form and LM is a midsegment of, so LM and LM = 1 2 JK is a midsegment of, so JK and JK = 1 2 This shows that LM JK and LM = JK, so LM JK pair of opposite sides of JKLM are both parallel and congruent, so JKLM is a parallelogram K Module Lesson Lesson 24 2

14 Lesson Performance Task In this lesson you ve learned three theorems for confirming that a figure is a parallelogram If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram For each of the following situations, choose one of the three theorems and use it in your explanation You should choose a different theorem for each explanation a You re an amateur astronomer, and one night you see what appears to be a parallelogram in the constellation of Lyra Explain how you could verify that the figure is a parallelogram ONNET VOULRY What is the converse of a statement? The converse is the statement that results from interchanging the hypothesis and conclusion of the given statement What is the converse of this statement: If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent? If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram b You have a frame shop and you want to make an interesting frame for an advertisement for your store You decide that you d like the frame to be a parallelogram but not a rectangle Explain how you could construct the frame c You re using a toolbox with cantilever shelves like the one shown here Explain how you can confirm that the brackets that attach the shelves to the box form a parallelogram a Possible answer: You could take a photograph of the figure and with a protractor measure to see if the opposite angles are congruent (If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram) b Possible answer: hoose two lengths of framing materials of one length and two of another (If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram) s long as the angles of the frame are not right angles, the frame is not a rectangle c Possible answer: ttach diagonal braces to opposite corners and check to see if they bisect each other (If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram) Houghton Mifflin Harcourt Publishing ompany INTEGRTE MTHEMTIL PRTIES Focus on ritical Thinking MP3 Give an example of a theorem involving quadrilaterals that is true but whose converse is false Explain why the converse is false Sample answer: Theorem: If a figure is a square, then it has four sides onverse: If a figure has four sides, then it is a square The converse is false because there are many four-sided figures that are not squares, including trapezoids and rhombuses that contain no right angles Module Lesson 2 EXTENSION TIVITY The night sky is divided into 88 constellations, groups of stars that appear to form figures or shapes Many of the constellations are wholly or partially geometrical in design Have students research the constellations, looking for triangles and quadrilaterals in the shapes Students may wish to start with the parallelogram in Lyra, shown in the photo in the Lesson Performance Task Preview Students may draw or print the shapes they find, and add to them interesting information about the shapes, such as the names of the stars that form them, the distances of the stars, and traditional stories told about the constellations in which the shapes appear Scoring Rubric 2 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 0 points: Student does not demonstrate understanding of the problem onditions for Parallelograms 1216