Properties of Parallelograms 6-2 Properties of Parallelograms Warm Up Lesson Presentation Lesson Quiz Holt Geometry
Are you ready? Find the value of each variable. 1. x 2. y 3. z
Objectives TSW prove and apply properties of parallelograms. TSW use properties of parallelograms to solve problems.
Vocabulary parallelogram
Any polygon with four sides is a quadrilateral Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.
Opposite sides of a quadrilateral do not share a vertex Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side. Helpful Hint
A quadrilateral with two pairs of parallel sides is a parallelogram A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .
Example 1: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find CF.
Example 2: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°.
Example 3: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find DF.
Example 4 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find KN.
Example 5 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find mNML. Def. of angles.
Example 6 In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find LO.
Example 7: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ.
Example 8: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find mZ .
Example 9 EFGH is a parallelogram. Find JG.
Example 10 EFGH is a parallelogram. Find FH.
When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices. Remember!
Example 11: Parallelograms in the Coordinate Plane Three vertices of JKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M. Since JKLM is a parallelogram, both pairs of opposite sides must be parallel. J K L
Example 12 Three vertices of PQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R. Since PQRS is a parallelogram, both pairs of opposite sides must be parallel. P Q S
6.3 Are you ready? 1. 2. 3. 2x + 7 4. 16x – 9 5. (8y + 5)° Justify each statement. 1. 2. Evaluate each expression for x = 12 and y = 8.5. 3. 2x + 7 4. 16x – 9 5. (8y + 5)°
Objective TSW prove that a given quadrilateral is a parallelogram.
You have learned to identify the properties of a parallelogram 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 can use the definition of a parallelogram or the conditions below.
The two theorems below can also be used to show that a given quadrilateral is a parallelogram.
Example 1: Verifying Figures are Parallelograms Show that JKLM is a parallelogram for a = 3 and b = 9.
Example 2: Verifying Figures are Parallelograms Show that PQRS is a parallelogram for x = 10 and y = 6.5.
Example 3 Show that PQRS is a parallelogram for a = 2.4 and b = 9.
Example 4: Applying Conditions for Parallelograms Determine if the quadrilateral must be a parallelogram. Justify your answer.
Example 5: Applying Conditions for Parallelograms Determine if the quadrilateral must be a parallelogram. Justify your answer.
Example 6 Determine if the quadrilateral must be a parallelogram. Justify your answer.
Example 7 Determine if each quadrilateral must be a parallelogram. Justify your answer.
To say that a quadrilateral is a parallelogram by definition, you must show that both pairs of opposite sides are parallel. Helpful Hint
Example 8: Proving Parallelograms in the Coordinate Plane Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0).
Example 9: Proving Parallelograms in the Coordinate Plane Show that quadrilateral ABCD is a parallelogram by using Theorem 6-3-1. A(2, 3), B(6, 2), C(5, 0), D(1, 1).
Example 10 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.
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.
To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions. Helpful Hint
Example 11: Application The 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?
Example 12 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?
Lesson Quiz: Part I 1. 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. 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 II 3. 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.
Check It Out! Example 3 Continued Step 4 Use the slope formula to verify that P Q S R The coordinates of vertex R are (7, 6).
Lesson Quiz: Part I In PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure. 1. PW 2. mPNW 18 144°
Lesson Quiz: Part II QRST is a parallelogram. Find each measure. 2. TQ 3. mT 71° 28
Lesson Quiz: Part III 5. Three vertices of ABCD are A (2, –6), B (–1, 2), and C(5, 3). Find the coordinates of vertex D. (8, –5)
Lesson Quiz: Part IV 6. Write a two-column proof. Given: RSTU is a parallelogram. Prove: ∆RSU ∆TUS Statements Reasons 1. RSTU is a parallelogram. 1. Given 4. SAS 4. ∆RSU ∆TUS 3. R T 2. cons. s 3. opp. s