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Properties of Parallelograms Chapter 6 - Lesson 2 Properties of Parallelograms

Objective: The students will be able to prove and apply properties of parallelograms to solve problems.

Any polygon with 4 sides is a quadrilateral Any polygon with 4 sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names. Quadrilateral

Helpful Hints Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.

Definition Parallelogram Figure ABCD is a Parallelogram a quadrilateral with both pairs of opposite sides parallel Figure ABCD is a Parallelogram

Example 1A: Properties of Parallelograms In CDEF, DE = 74 mm, Find CF.  opp. sides  CF = DE Def. of  segs. CF = 74 mm Substitute 74 for DE.

Check It Out! Example 1a In KLMN, LM = 28 in. Find KN.  opp. sides  LM = KN Def. of  segs. LM = 28 in. Substitute 28 for DE.

Example 1B: Properties of Parallelograms In CDEF, mFCD = 42°. Find mEFC. mFCD + mEFC = 180°  consecutive angles are supp. 42° + mEFC = 180° Substitution mEFC = 138° Subtraction

Check It Out! Example 1b In KLMN, mLKN = 74°. Find mNML. NML  LKN  opp. s  mNML = mLKN Def. of  s. mNML = 74° Substitute 74° for mLKN.

Example 1C: Properties of Parallelograms In CDEF, DG = 31. Find DF. DG = GF  diagonals bisect each other GF = 31 Substitution DF = DG + GF Segment Addition Postulate DF = 31 + 31 Substitution DF = 62 Simplify

Check It Out! Example 1c In KLMN, LN = 26 in. Find LO. LN = 2LO  diags. bisect each other. 26 = 2LO Substitute 26 for LN. LO = 13 in. Simplify.

Example 1D: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ.  opposite sides are  XW = YZ  XW = YZ Definition of  segments 8a – 4 = 6a + 10 Substitution YZ = 8a - 4 Given (diagram) 2a – 4 = 10 Subtraction YZ = 8(7) - 4 Substitution 2a = 14 Addition YZ = 56 - 4 Simplify a = 7 Division YZ = 52 Simplify

Check It Out! Example 1d EFGH is a parallelogram. Find JG. EJ = JG  diagonals bisect each other 3w = w + 8 Substitution JG = w + 8 Given 2w = 8 Subtraction JG = 4 + 8 Substitution w = 4 Division JG = 12 Simplify

RSTU is a parallelogram. Find and y. 40 58 Example 2-2a

ABCD is a parallelogram. Answer: Example 2-2d

Example 3: 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 Step 1 Graph the given points.

Step 2 Find the slope of by counting the units from K to L. Example 3 Continued Step 2 Find the slope of by counting the units from K to L. The rise from 2 to 6 is 4. The run of –2 to 2 is 4. J K L Step 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. M Answer: The coordinates of vertex M are (7, –4).

Answer: The coordinates of vertex R are (3, –6). 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. P Q S Down 6 Down 6 Over 2 Over 2 R Answer: The coordinates of vertex R are (3, –6).

Example 4: Using Properties of Parallelograms in a Proof Write a two-column proof. Given: ABCD is a parallelogram. Prove: ∆AEB  ∆CED Statements Reasons 4. 1. ABCD is a parallelogram 1. Given 2.  opposite sides  3.  diagonals bisect each other ∆ ∆ 4. SSS

Write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: H M Statements Reasons 1. GHJN and JKLM are parallelograms. 1. Given 2. H and HJN are supp. M and MJK are supp. 2.  cons. s supp. 3. HJN  MJK 3. Vert. s Theorem 4. H  M 4.  Supplements Theorem

Opposite sides of a parallelogram are . Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent. Given: Prove: Proof: Reasons Statements 1. Given 2. Given 3. Opposite sides of a parallelogram are . 4. 4. Substitution Property Example 2-1a

Prove that if and are the diagonals of , . Given: Prove: Proof: Reasons Statements 1. 1. Given 2. 2. Opposite sides of a parallelogram are congruent. 3. CBD 3. If 2 lines are cut by a transversal, alternate interior s are . 4. 4. Angle-Side-Angle Example 2-1c

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Lesson Summary: Objective: The students will be able to prove and apply properties of parallelograms to solve problems.

Preview of the Next Lesson: Objective: The students will prove quadrilaterals parallelograms.

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