# 6-3 Conditions for Parallelograms Holt Geometry Warm Up Warm Up Lesson Presentation Lesson...

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6-3Conditions for ParallelogramsHolt GeometryWarm UpLesson PresentationLesson Quiz

Warm UpJustify each statement.

1. 2. Evaluate each expression for x = 12 and y = 8.5.

3. 2x + 74. 16x 95. (8y + 5)Reflex Prop. of Conv. of Alt. Int. s Thm.31183736.3 Properties of Parallelograms

Prove that a given quadrilateral is a parallelogram.Objective6.3 Properties of Parallelograms

You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you canuse the definition of a parallelogram or the conditions below.6.3 Properties of Parallelograms

6.3 Properties of Parallelograms

The two theorems below can also be used to show that a given quadrilateral is a parallelogram.6.3 Properties of Parallelograms

Example 1A: Verifying Figures are ParallelogramsShow that JKLM is a parallelogram for a = 3 and b = 9.Step 1 Find JK and LM.Given Substitute and simplify. JK = 15a 11 JK = 15(3) 11 = 34 LM = 10a + 4 LM = 10(3)+ 4 = 34 6.3 Properties of Parallelograms

Example 1A ContinuedSince JK = LM and KL = JM, JKLM is a parallelogram by Theorem 6-3-2.Step 2 Find KL and JM.Given Substitute and simplify. KL = 5b + 6 KL = 5(9) + 6 = 51JM = 8b 21 JM = 8(9) 21 = 516.3 Properties of Parallelograms

Example 1B: Verifying Figures are ParallelogramsShow that PQRS is a parallelogram for x = 10 and y = 6.5.Given Substitute 6.5 for y and simplify.Given Substitute 6.5 for y and simplify.mQ = (6y + 7)mQ = [(6(6.5) + 7)] = 46 mS = (8y 6)mS = [(8(6.5) 6)] = 46 mR = (15x 16)mR = [(15(10) 16)] = 134 Given Substitute 10 for x and simplify.6.3 Properties of Parallelograms

Example 1B ContinuedSince 46 + 134 = 180, R is supplementary to both Q and S. PQRS is a parallelogram by Theorem 6-3-4.6.3 Properties of Parallelograms

Check It Out! Example 1 Show that PQRS is a parallelogram for a = 2.4 and b = 9.By Theorem 6-3-1, PQRS is a parallelogram.mQ = 74, and mR = 106, so Q and R are supplementary. So one pair of opposite sides of PQRS are || and .6.3 Properties of Parallelograms

Example 2A: Applying Conditions for ParallelogramsDetermine if the quadrilateral must be a parallelogram. Justify your answer.Yes. The 73 angle is supplementary to both its corresponding angles. By Theorem 6-3-4, the quadrilateral is a parallelogram.6.3 Properties of Parallelograms

Example 2B: Applying Conditions for ParallelogramsDetermine if the quadrilateral must be a parallelogram. Justify your answer.No. One pair of opposite angles are congruent. The other pair is not. The conditions for a parallelogram are not met. 6.3 Properties of Parallelograms

Check It Out! Example 2a Determine if the quadrilateral must be a parallelogram. Justify your answer.The diagonal of the quadrilateral forms 2 triangles.YesTwo angles of one triangle are congruent to two angles of the other triangle, so the third pair of angles are congruent by the Third Angles Theorem.So both pairs of opposite angles of the quadrilateral are congruent .By Theorem 6-3-3, the quadrilateral is a parallelogram.6.3 Properties of Parallelograms

Check It Out! Example 2b Determine if each quadrilateral must be a parallelogram. Justify your answer.No. Two pairs of consective sides are congruent.None of the sets of conditions for a parallelogram are met.6.3 Properties of Parallelograms

6.3 Properties of Parallelograms

Example 3: Parallelograms in the Coordinate PlaneThree vertices of JKLM are J(3, 8), K(2, 2), and L(2, 6). Find the coordinates of vertex M.Step 1 Graph the given points.Since JKLM is a parallelogram, both pairs of opposite sides must be parallel.6.3 Properties of Parallelograms

Example 3 ContinuedStep 3 Start at J and count the same number of units.A rise of 4 from 8 is 4.A run of 4 from 3 is 7. Label (7, 4) as vertex M.M6.3 Properties of Parallelograms

The coordinates of vertex M are (7, 4).Example 3 Continued6.3 Properties of Parallelograms

Check It Out! Example 3 Three vertices of PQRS are P(3, 2), Q(1, 4), and S(5, 0). Find the coordinates of vertex R.Step 1 Graph the given points.Since PQRS is a parallelogram, both pairs of opposite sides must be parallel.6.3 Properties of Parallelograms

Step 3 Start at S and count the same number of units.A rise of 6 from 0 is 6.A run of 2 from 5 is 7. Label (7, 6) as vertex R.Check It Out! Example 3 ContinuedR6.2 Properties of Parallelograms

Check It Out! Example 3 ContinuedThe coordinates of vertex R are (7, 6).6.2 Properties of Parallelograms

Example 3A: Proving Parallelograms in the Coordinate PlaneShow that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(1, 6), K(4, 1), L(4, 5), M(7, 0).Find the slopes of both pairs of opposite sides.Since both pairs of opposite sides are parallel, JKLM is a parallelogram by definition.6.3 Properties of Parallelograms

Example 3B: Proving Parallelograms in the Coordinate PlaneShow that quadrilateral ABCD is a parallelogram by using Theorem 6-3-1. A(2, 3), B(6, 2), C(5, 0), D(1, 1).Find the slopes and lengths of one pair of opposite sides.6.3 Properties of Parallelograms

Check It Out! Example 3 Use the definition of a parallelogram to show that the quadrilateral with vertices K(3, 0), L(5, 7), M(3, 5), and N(5, 2) is a parallelogram.6.3 Properties of Parallelograms

You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply.6.3 Properties of Parallelograms

6.3 Properties of Parallelograms

Example 4: ApplicationThe legs of a keyboard tray are connected by a bolt at their midpoints, which allows the tray to be raised or lowered. Why is PQRS always a parallelogram?Since the bolt is at the midpoint of both legs, PE = ER and SE = EQ. So the diagonals of PQRS bisect each other, and by Theorem 6-3-5, PQRS is always a parallelogram.6.3 Properties of Parallelograms

Check It Out! Example 4 The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same?Therefore the viewing never changes.6.3 Properties of Parallelograms

No; One pair of consecutive s are , and one pair of opposite sides are ||. The conditions for a parallelogram are not met.Lesson Quiz: Part I1. Show that JKLM is a parallelogram for a = 4 and b = 5.

2. Determine if QWRT must be a parallelogram. Justify your answer.

JN = LN = 22; KN = MN = 10; so JKLM is a parallelogram by Theorem 6-3-5.6.3 Properties of Parallelograms

Lesson Quiz: Part II3. Show that the quadrilateral with vertices E(1, 5), F(2, 4), G(0, 3), and H(3, 2) is a parallelogram.Since one pair of opposite sides are || and , EFGH is a parallelogram by Theorem 6-3-1.6.3 Properties of Parallelograms

Lesson Quiz: Part III4. Three vertices of ABCD are A (2, 6), B (1, 2), and C(5, 3). Find the coordinates of vertex D.(8, 5)6.3 Properties of Parallelograms

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