6-2. Properties of Parallelograms. Focus on Reasoning INTR O D U C E. Standards for Mathematical Content 1. Investigate parallelograms.

Transcription

1 6-2 Properties of Parallelograms Focus on Reasoning Essential question: What can you conclude about the sides, angles, and diagonals of a parallelogram? TEH Standards for Mathematical ontent 1 G-O.3.11 Prove theorems about parallelograms. G-SRT.2.5 Use congruence... criteria for triangles to solve problems and to prove relationships in geometric figures. Materials: geometry software Questioning Strategies s you use the software to drag points,,, and/or, does the quadrilateral remain a parallelogram? Why? Yes; the lines Vocabulary diagonal that form opposite sides remain parallel. What do you notice about consecutive angles in the parallelogram? Why does this make sense? onsecutive angles are supplementary. Prerequisites Theorems about parallel lines cut by a transversal Triangle congruence criteria This makes sense because opposite sides are parallel, so consecutive angles are same-side interior angles. y the Same-Side Interior ngles Postulate, these angles are supplementary. Math ackground In this lesson, students extend their earlier work with triangle congruence criteria and triangle properties to prove facts about parallelograms. This lesson gives students a chance to use inductive and deductive reasoning to investigate properties of the sides, angles, and diagonals of parallelograms. Students have encountered parallelograms in earlier grades. sk a volunteer to define parallelogram. Students may have only an informal idea of what a parallelogram is (e.g., a slanted rectangle ), so be sure they understand that the mathematical definition of a parallelogram is a quadrilateral with two pairs of parallel sides. You may want to show students how they can make a parallelogram by drawing lines on either side of a ruler, changing the position of the ruler, and drawing another pair of lines. sk students to explain why this method creates a parallelogram. 2 Prove that opposite sides of a parallelogram are congruent. Questioning Strategies Why do you think the proof is based on drawing the diagonal? rawing the diagonal creates two triangles; then you can use triangle congruence criteria and PT. 245 Lesson 2 Teaching Strategy Some students may have difficulty with terms like opposite sides or consecutive angles. Remind students that opposite sides of a quadrilateral do not share a vertex (that is, they do not intersect). onsecutive sides of a quadrilateral do share a vertex (that is, they intersect). Opposite angles of a quadrilateral do not share a side. onsecutive angles of a quadrilateral do share a side. You may want to help students draw and label a quadrilateral for reference. INTR O U E hapter 6 Investigate parallelograms.

2 Name lass ate Properties of Parallelograms Focus on Reasoning Essential question: What can you conclude about the sides, angles, and diagonals of a parallelogram? Recall that a parallelogram is a quadrilateral that has two pairs of parallel sides. You use the symbol to name a parallelogram. For example, the figure shows. G-O.3.11, G-SRT Notes 1 Investigate parallelograms. Use the straightedge tool of your geometry software to draw a straight line. Then plot a point that is not on the line. Select the point and line, go to the onstruct menu, and construct a line through the point that is parallel to the line. This will give you a pair of parallel lines, as shown. Repeat Step to construct a second pair of parallel lines that intersect those from Step. The intersections of the parallel lines create a parallelogram. Plot points at these intersections. Label the points,,, and. Use the Measure menu to measure each angle of the parallelogram. E Use the Measure menu to measure the length of each side of the parallelogram. (You can do this by measuring the distance between consecutive vertices.) F rag the points and lines in your construction to change the shape of the parallelogram. s you do so, look for relationships in the measurements. = 2.60 cm = 1.74 cm = 2.60 cm = 1.74 cm m = m = m = m = REFLET 1a. Make a conjecture about the sides and angles of a parallelogram. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. hapter Lesson 2 You may have discovered the following theorem about parallelograms. Theorem If a quadrilateral is a parallelogram, then opposite sides are congruent. 2 Prove that opposite sides of a parallelogram are congruent. omplete the proof. Given: is a parallelogram. Prove: _ and Statements Reasons 1. is a parallelogram. 1. Given 2. raw. 2. Through any two points there exists exactly one line. 3. ǁ ; ǁ 3. efinition of parallelogram 4. ; 4. lternate Interior ngles Theorem Reflexive Property of ongruence S ongruence riterion 7. ; 7. PT REFLET 2a. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem. Under a 180 rotation about the center of the parallelogram, each side is mapped to its opposite side. Since rotations preserve distance, this shows that opposite sides are congruent. 2b. One side of a parallelogram is twice as long as another side. The perimeter of the parallelogram is 24 inches. Is it possible to find all the side lengths of the parallelogram? If so, find the lengths. If not, explain why not. Yes; consecutive sides have lengths x, 2x, x, and 2x, so x + 2x + x + 2x = 24, or 6x = 24. Therefore x = 4 and the side lengths are 4 in., 8 in., 4 in., and 8 in. hapter Lesson 2 hapter Lesson 2

3 TEH Investigate diagonals of 3 parallelograms. Materials: geometry software Questioning Strategies How many diagonals does a parallelogram have? Is this true for every quadrilateral? Two; yes If a quadrilateral is named PQRS, what are the diagonals? PR and QS re the diagonals of a parallelogram ever congruent? If so, when does this appear to happen? Yes; when the parallelogram is a rectangle Prove diagonals of a parallelogram 4 bisect each other. Questioning Strategies Why do you think this theorem was introduced after the theorems about the sides and angles of a parallelogram? The proof of this theorem depends upon the fact that opposite sides of a parallelogram are congruent. Highlighting the Standards s students work on the proof in this lesson, ask them to think about how the format of the proof makes it easier to understand the underlying structure of the argument. This addresses elements of Standard 3 (onstruct viable arguments and critique the reasoning of others). Students should recognize that a flow proof shows how one statement connects to the next. This may not be as apparent in a two-column format. You may want to have students rewrite the proof in a two-column format as a way of exploring this further. hapter Lesson 2

4 Essential question: What can you conclude about the diagonals of a parallelogram? segment that connects any two nonconsecutive vertices of a polygon is a diagonal. parallelogram has two diagonals. In the figure, and are diagonals of. Notes 3 Investigate diagonals of parallelograms. Use geometry software to construct a parallelogram. (See Lesson 4-2 for detailed instructions.) Label the vertices of the parallelogram,,, and. Use the segment tool to construct the diagonals, and. Plot a point at the intersection of the diagonals. Label this point E. Use the Measure menu to measure the length of E, E, E, and E. (You can do this by measuring the distance between the relevant endpoints.) E rag the points and lines in your construction to change the shape of the parallelogram. s you do so, look for relationships in the measurements. E E = 1.52 cm E = 2.60 cm E = 1.52 cm E = 2.60 cm REFLET 3a. Make a conjecture about the diagonals of a parallelogram. The diagonals of a parallelogram bisect each other. 3b. student claims that the perimeter of E is always equal to the perimeter of E. Without doing any further measurements in your construction, explain whether or not you agree with the student s statement. gree; E = E, E = E, and = since opposite sides of a parallelogram are congruent. So, E + E + = E + E +. hapter Lesson 2 You may have discovered the following theorem about parallelograms. Theorem If a quadrilateral is a parallelogram, then the diagonals bisect each other. 4 Prove diagonals of a parallelogram bisect each other. omplete the proof. Given: is a parallelogram. Prove: E E and E E. E E lternate Interior ngles Theorem REFLET efinition of parallelogram is a parallelogram. Given E E lternate Interior ngles Theorem 4a. Explain how you can prove the theorem using a different congruence criterion. E E S ongruence riterion E E and E E. PT E Opposite sides of a parallelogram are congruent. E E because they are vertical angles. Using this fact plus the fact that E E and, it is possible to prove the theorem using the S ongruence riterion. hapter Lesson 2 hapter Lesson 2

5 Teaching Strategy The lesson concludes with the theorem that states that opposite angles of a parallelogram are congruent. The proof of this theorem is left as an exercise (Exercise 1). e sure students recognize that the proof of this theorem is similar to the proof that opposite sides of a parallelogram are congruent. Noticing such similarities is an important problem-solving skill. Highlighting the Standards Exercise 4 is a multi-part exercise that includes opportunities for mathematical modeling, reasoning, and communication. It is a good opportunity to address Standard 4 (Model with mathematics). raw students attention to the way they interpret their mathematical results in the context of the real-world situation. Specifically, ask students to explain what their mathematical findings tell them about the appearance and layout of the park. LOSE Essential Question What can you conclude about the sides, angles, and diagonals of a parallelogram? Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. Summarize Have students make a graphic organizer to summarize what they know about the sides, angles, and diagonals of a parallelogram. sample is shown below. PRTIE Exercise 1: Students practice what they learned in part 2 of the lesson. Exercise 2: Students use reasoning to extend what they know about parallelograms. Exercise 3: Students use reasoning and/or algebra to find unknown angle measures. Exercise 4: Students apply their learning to solve a multi-step real-world problem. Parallelogram Opposite sides are congruent. The diagonals bisect each other. If E is the point where diagonals and intersect, then E E and E E. Opposite angles are congruent. hapter Lesson 2

6 The angles of a parallelogram also have an important property. It is stated in the following theorem, which you will prove as an exercise. Notes Theorem If a quadrilateral is a parallelogram, then opposite angles are congruent. PRTIE 1. Prove the above theorem about opposite angles of a parallelogram. Given: is a parallelogram. Prove: and (Hint: You only need to prove that. similar argument can be used to prove that. lso, you may or may not need to use all the rows of the table in your proof.) Statements Reasons 1. is a parallelogram. 1. Given 2. raw Through any two points there. 2. exists exactly one line. 3. ; ; efinition of parallelogram 2. Explain why consecutive angles of a parallelogram are supplementary. lternate Interior ngles Theorem Reflexive Property of ongruence S ongruence riterion PT onsecutive angles of a parallelogram are same-side interior angles for a pair of parallel lines (the opposite sides of the parallelogram), so the angles are supplementary by the Same-Side Interior ngles Postulate. 3. In the figure, JKLM is a parallelogram. Find the measure of each of the numbered angles. J K N m 1 = 19 ; m 2 = 43 ; m 3 = 118 ; m 4 = 118 ; m 5 = M L hapter Lesson 2 4. city planner is designing a park in the shape of a parallelogram. s shown in the figure, there will be two straight paths through which visitors may enter the park. The paths are bisectors of consecutive angles of the parallelogram, and the paths intersect at point P. P a. Work directly on the parallelograms below and use a compass and straightedge to construct the bisectors of and. Then use a protractor to measure P in each case. P Make a conjecture about P. P is a right angle. b. Write a paragraph proof to show that your conjecture is always true. (Hint: Suppose m P = x, m P = y, and m P = z. What do you know about x + y + z? What do you know about 2x + 2y?) y the Triangle Sum Theorem, x + y + z = 180. lso, m = (2x) and m = (2y). y the Same-Side Interior ngles Postulate m + m = 180. So 2x + 2y = 180 and x + y = 90. Substituting this in the first equation gives 90 + z = 180 and z = 90. c. When the city planner takes into account the dimensions of the park, she finds that point P lies on x y x y, as shown. Explain why it must be the case that = 2. (Hint: Use congruent base angles to show z that P and P are isosceles.) P P P since P is an angle bisector. lso, P P by the lternate Interior ngles Theorem. Therefore, P P. This means P is isosceles, with P. Similarly, P. lso, as opposite sides of a parallelogram. So, = P + P = + = + P = 2. hapter Lesson 2 hapter Lesson 2

7 ITIONL PRTIE N PROLEM SOLVING ssign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition. nswers dditional Practice cm cm cm (0, -3) 15. Possible answer: Statements Problem Solving 1. m = 135 ; m = 45 Reasons 1. EFG is a parallelogram. 1. Given 2. m EG = m EH + 2. ngle dd. Post. m GH, m FG = m FGH + m GH 3. m EG + m FG = cons. s supp. 4. m EH + m GH + m FGH + m GH = m GH + m GH + m HG = m GH + m GH + m HG = m EH + m GH + m FGH + m GH 7. m HG = m EH + m FGH 4. Subst. (Steps 2, 3) 5. Triangle Sum Thm. 6. Trans. Prop. of = 7. Subtr. Prop. of = in ft H 7. hapter Lesson 2

8 Name lass ate dditional Practice 6-2 Notes hapter Lesson 2 Problem Solving hapter Lesson 2 hapter Lesson 2