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6.2 Parallelograms
Objectives
Use Relationships Among Sides and Angles of Parallelograms.
Use Relationships Among Consecutive Angles and Diagonals of Parallelograms.
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Definition
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Parallelograms have special properties regarding their sides, angles, and diagonals.
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Definition
In a parallelogram (and in all other convex quadrilaterals), opposite sides do not share a vertex, and opposite angles do not share a side.
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4. Opposite Sides of a Parallelogram
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5. Opposite Angles of a Parallelogram
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Proof
Use the given figure for a proof of Theorem Given: ▱ABCD Prove: AB ≅ CD and BC ≅ DA Solution Start a two-column proof. This is on the next slide.
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Statements
Reasons
1. ABCD is a parallelogram.
1. Given
2. Definition of parallelogram
3. If lines are parallel, then alternate interior angles are congruent.
4. Reflexive Property of Congruence
5. ASA
6. Corresponding parts of congruent triangles are congruent. (cpoctac)
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Definition
Angles of a polygon that share a side are consecutive angles. In the diagram, ∠A and ∠B are consecutive angles because they share side AB.
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9. Consecutive Angles of a Parallelogram
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10. Diagonals of a Parallelogram
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11. Using Consecutive Angles
Multiple Choice What is m∠P in ▱PQRS? a. 26° b. 64° c. 116° d. 126° Solution ▱PQRS means parallelogram PQRS
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12. Using Consecutive Angles
Multiple Choice What is m∠P in ▱PQRS? a. 26° b. 64° c. 116° d. 126° Solution m∠P + m∠S = 180° m∠P + 64° = 180° m∠P = 116° The correct answer is c.
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13. Using Algebra to Find Lengths
Solve a system of linear equations to find the values of x and y in ▱KLMN. What are KM and LN? Solution To find these lengths, we set up a system of two equations and solve by substitution.
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14. Using Algebra to Find Lengths
Equation (1) y + 10 = 2x – 8 Equation (2) x = y + 2 y + 10 = 2x − 8 y + 10 = 2(y + 2) − 8 y + 10 = 2y + 4 − 8 y + 10 = 2y − 4 10 = y − 4 14 = y
To find the value of x, choose Equation 1 or Equation 2 and substitute 14 for y. We
choose Equation 2.
x = y + 2
x =
= 16
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15. Using Algebra to Find Lengths
Now, let’s find KM and LN. By looking at the parallelogram, we see that KM = 2(KP) and LN = 2(LP). KM = 2(KP) = 2(y + 10) = 2( ) = 48 Thus, x = 16 and y = 14. Also, KM = 48 units and LN = 32 units.
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16. Multiple Parallel Lines and Transversals
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17. Using Parallel Lines and Transversals
In the figure, AB = BC = CD = 2 and EF = What is EH? Solution We are given the length of segment EF. Thus, to find EH, we need the length of segments FG and GH.
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