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Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems.

Explore various examples and proofs showcasing the properties of parallelograms, including finding measures, coordinates of vertices, and solving problems in the coordinate plane.

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Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems.

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  1. Objectives Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems.

  2. _________________ - A quadrilateral with two pairs of parallel sides.

  3. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find CF. Example 1A: Properties of Parallelograms

  4. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find mEFC. Example 1B: Properties of Parallelograms

  5. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find DF. Example 1C: Properties of Parallelograms

  6. Example 2A: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find mZ.

  7. Example 2B EFGH is a parallelogram. Find JG.

  8. L K J 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. Step 1 Graph the given points.

  9. Step 2 Find the slope of by counting the units from K to L. L K J Example 3 Continued Step 3 Start at J and count the same number of units. M

  10. Step 4 Use the slope formula to verify that L K M J Example 3 Continued

  11. Q S P Example 3B 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. Step 1 Graph the given points.

  12. Step 2 Find the slope of by counting the units from P to Q. Q S P Example 3B Continued R Step 3 Start at S and count the same number of units.

  13. Step 4 Use the slope formula to verify that R Q S P Example 3B Continued

  14. Example 4A: Using Properties of Parallelograms in a Proof Write a two-column proof. Given: ABCD is a parallelogram. Prove:∆AEB∆CED

  15. Example 4A Continued Proof: 1. 1. 2. 3. 4.

  16. Example 4B: Using Properties of Parallelograms in a Proof Write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove:H M

  17. 2. Example 4B Continued Proof: 1. 1. 2. 3. 3. 4. 4.

  18. Example 4C Write a two-column proof. Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear. Prove: N  K

  19. 2. Example 4C Continued Proof: 1. 1. 2. 3. 3. 4. 4.

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