Proving That a Quadrilateral Is a Parallelogram. To determine whether a quadrilateral is a parallelogram

Transcription

1 - roving That a Quadrilateral Is a arallelogram ontent Standards G.O. rove theorems about parallelograms... the diagonals of a parallelogram bisect each other and its converse... lso G.ST. Objective To determine whether a quadrilateral is a parallelogram an you visualize parallelograms composed of triangles in the pattern? ach section of glass in the exterior of a building in Macau, hina, forms an equilateral triangle. o you think the window washer s feet stay parallel to the ground as he lands at each level of windows? xplain. (ssume that the bases of the lowest triangles are parallel to the ground.) T YNMI I V I S S I T MTHMTIL TIS ynamic ctivity arallelogram onditions In the Solve It, you used angle properties to show that lines are parallel. In this lesson, you will apply the same properties to show that a quadrilateral is a parallelogram. ssential Understanding You can decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties. In Lesson -, you learned theorems about the properties of parallelograms. In this lesson, you will learn the converses of those theorems. That is, if a quadrilateral has certain properties, then it must be a parallelogram. - is the converse of -. - If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. is a You will prove - in xercise. s - and - are the converses of s - and -, respectively. They use angle relationships to conclude that a quadrilateral is a parallelogram. Lesson - roving That a Quadrilateral Is a arallelogram

2 - If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. m m m m is a You will prove - in xercise. - If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. is a You will prove - in xercise. You can use algebra together with s -, -, and - to find segment lengths and angle measures that assume that a quadrilateral is a parallelogram. roblem Finding Values for arallelograms Which theorem should you use? The diagram gives you information about sides. Use - because it uses sides to conclude that a quadrilateral is a parallelogram. For what value of y must QS be a parallelogram? Step Find x. Step Find y. x x If opp. sides are, then the quad. is a. x Subtract x from each side. y x x dd to each side. If opp. sides are, then the quad. is a. Substitute for x. Simplify. For QS to be a parallelogram, the value of y must be. x S x x Q y Got It?. Use the diagram at the right. For what values of x and y must FGH be a parallelogram? (y ) (x ) F H (y ) (x ) G hapter olygons and Quadrilaterals

3 You know that the converses of s -, -, and - are true. Using what you have learned, you can show that the converse of - is also true. - If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. is a roof roof of - Given: and bisect each other at. rove: is a parallelogram. and bisect each other at. Given / > / Vertical? are >. > > ef. of segment bisector / > / Vertical? are >. n > SS > SS / > / / > / orresp. parts of > > are >. orresp. parts of > > are >. i If alternate interior? >, then lines are i. i If alternate interior? >, then lines are i. is a parallelogram. ef. of parallelogram Lesson - roving That a Quadrilateral Is a arallelogram

4 - suggests that if you keep two objects of the same length parallel, such as cross-country skis, then the quadrilateral formed by connecting their endpoints is always a parallelogram. - If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. is a You will prove - in xercise. How do you decide if you have enough information? If you can satisfy every condition of a theorem about parallelograms, then you have enough information. roblem eciding Whether a Quadrilateral Is a arallelogram an you prove that the quadrilateral is a parallelogram based on the given information? xplain. Given:,, Given: HI HK, JI JK m, m rove: HIJK is a parallelogram. rove: is a parallelogram. H I K J Yes. Same-side interior angles and are supplementary, so. Since, is a parallelogram by -. No. y -, you need to show that both pairs of opposite sides are congruent, not consecutive sides. Got It?. an you prove that the quadrilateral is a parallelogram based on the given information? xplain. a. Given: F G, FG b. Given: LN NL, NL LN rove: FG is a parallelogram. rove: LN is a parallelogram. N G F L hapter olygons and Quadrilaterals

5 s the arms of the lift move, what changes and what stays the same? The angles the arms form with the ground and the platform change, but the lengths of the arms and the platform stay the same. roblem Identifying arallelograms Vehicle Lifts truck sits on the platform of a vehicle lift. Two moving arms raise the platform until a mechanic can fit underneath. Why will the truck always remain parallel to the ground as it is lifted? xplain. Q Q ft ft ft ft ft ft ft S ft S The angles of QS change as platform Q rises, but its side lengths remain the same. oth pairs of opposite sides are congruent, so QS is a parallelogram by -. y the definition of a parallelogram, S Q. Since the base of the lift S lies along the ground, platform Q, and therefore the truck, will always be parallel to the ground. Got It?. easoning What is the maximum height that the vehicle lift can elevate the truck? xplain. oncept Summary roving That a Quadrilateral Is a arallelogram Method rove that both pairs of opposite sides are parallel. rove that both pairs of opposite sides are congruent. Source efinition of parallelogram - iagram rove that an angle is supplementary to both of its consecutive angles. - rove that both pairs of opposite angles are congruent. - rove that the diagonals bisect each other. - rove that one pair of opposite sides is congruent and parallel. - Lesson - roving That a Quadrilateral Is a arallelogram

6 Lesson heck o you know HOW?. For what value of y must LMN be a parallelogram? For xercises and, is the given information enough to prove that is a parallelogram? xplain... y x x y L y H M N o you UNSTN? MTHMTIL TIS. Vocabulary xplain why you can now write a biconditional statement regarding opposite sides of a parallelogram.. ompare and ontrast How is - in this lesson different from - in the previous lesson? In what situations should you use each theorem? xplain.. rror nalysis Your friend says, If a quadrilateral has a pair of opposite sides that are congruent and a pair of opposite sides that are parallel, then it is a parallelogram. What is your friend s error? xplain. ractice and roblem-solving xercises MTHMTIL TIS ractice lgebra For what values of x and y must be a parallelogram? See roblems and.. x.. x y x y x y (y ) (x ).. x. x x (x ) x (x ) an you prove that the quadrilateral is a parallelogram based on the given information? xplain..... Fishing Quadrilaterals are formed on the side of this fishing tackle box by the adjustable shelves and connecting pieces. xplain why the shelves are always parallel to each other no matter what their position is. See roblem. hapter olygons and Quadrilaterals

7 pply. Writing ombine each of s -, -, -, and - with its converse from this lesson into biconditional statements.. eveloping roof omplete this two-column proof of -. Given:, rove: is a parallelogram. Statements easons x y ) x y x y ) The sum of the measures of the angles of a quadrilateral is. ) (x y) ) a. ) x y ) b. ) and are supplementary. ) efinition of supplementary and are supplementary. ) c., ) d. ) is a parallelogram. ) e.. Think bout a lan rove -. roof Given:, rove: is a parallelogram. How can drawing diagonals help you? How can you use triangles in this proof?. rove -.. rove -. roof roof Given:, Given: is supplementary to. rove: is a parallelogram. is supplementary to. rove: is a parallelogram. lgebra For what values of the variables must be a parallelogram?.. y. (x ) (x ) x y y y (x ) (x ). Given: TS TW roof rove: STW is a parallelogram. S T. Open-nded Sketch two noncongruent parallelograms and FGH such that G and FH. W Lesson - roving That a Quadrilateral Is a arallelogram

8 hallenge. onstruction In the figure at the right, point is constructed by roof drawing two arcs. One has center and radius. The other has center and radius. rove that M is a median of.. robability If two opposite angles of a quadrilateral measure and the measures of the other angles are multiples of, what is the probability that the quadrilateral is a parallelogram? M Standardized Test rep ST/T. From which set of information can you conclude that STW is a parallelogram? S WT, S ST S ST, W WT S WT, ST W Z TZ, SZ WZ Short esponse xtended esponse. Write a proof using the diagram. Given: NJ T, JN T rove: JNT is a parallelogram.. Use the figure at the right. a. Write an equation and solve for x. b. Is F? xplain. c. Is F a parallelogram? xplain. N J (x ) F x (x ) T Mixed eview lgebra Find the value of each variable in each parallelogram... m. h e a (x ) h k ƒ x m. xplain how you can use overlapping congruent triangles to prove. Get eady! To prepare for Lesson -, do xercises. is a parallelogram and m. omplete the following m. m. m. m See Lesson -. c See Lessons - and -. See Lessons - and hapter olygons and Quadrilaterals