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Conditions for Parallelograms Chapter 6 - Lesson 3 Conditions for Parallelograms

Objective: The students will prove quadrilaterals parallelograms.

Theorems 6-3-1 to 6-3-5 can be used to show that a given quadrilateral is a parallelogram.

Example 1A: Verifying Figures are Parallelograms Verify that JKLM is a parallelogram for a = 3 and b = 9. If a = 3, then… If b = 9, then… JK = 15(3) - 11 KL = 5(9) + 6 JK = 34 KL = 51 LM = 10(3) + 4 JM = 8(9) – 21 LM = 34 JM = 51 LM = JK by Substitution JM = KL by Substitution LM  JK by Def.  segs. JM  KL by Def.  segs. Therefore JKLM is a parallelogram since both pairs of opposite sides are congruent.

Example 1B: Verifying Figures are Parallelograms Verify that PQRS is a parallelogram for x = 10 and y = 6.5. mQ = (6y + 7)° mQ = [(6(6.5) + 7)]° = 46° mS = (8y – 6)° mS = [(8(6.5) – 6)]° = 46° mR = (15x – 16)° mR = [(15(10) – 16)]° = 134° Since 46° + 134° = 180°, R is supplementary to both Q and S. Therefore PQRS is a parallelogram since an angle is supplementary to both of its consecutive angles.

Check It Out! Example 1 Verify that PQRS is a parallelogram for a = 2.4 and b = 9. PQ = RS = 16.8, so mQ = 74°, and mR = 106°, so Q and R are supplementary. Therefore, So one pair of opposite sides of PQRS are || and . Therefore PQRS is a parallelogram.

Check It Out! Example 2a Determine if the quadrilateral must be a parallelogram. Justify your answer. Yes The diagonal of the quadrilateral forms 2 triangles. Two 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.

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.

To say that a quadrilateral is a parallelogram by definition, you must show that both pairs of opposite sides are parallel. Helpful Hint

Example 3A: Proving Parallelograms in the Coordinate Plane Verify 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.

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. Helpful Hint To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions.

Example 4: 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? PE = ER and SE = EQ by definition of midpoint so the diagonals of PQRS bisect each other by the definition of segment bisector. Therefore PQRS is always a parallelogram because the diagonals bisect each other.

Prove: ABCD is a parallelogram. Write a paragraph proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram. Given: Prove: ABCD is a parallelogram. Proof: CPCTC. Therefore, ABCD is a parallelogram since both pairs of opposite sides of a quadrilateral are congruent. Example 3-1a

Prove: WXYZ is a parallelogram. Write a paragraph proof of the statement: If two diagonals of a quadrilateral divide the quadrilateral into four triangles where opposite triangles are congruent, then the quadrilateral is a parallelogram. Given: Prove: WXYZ is a parallelogram. Proof: by CPCTC. Therefore, WXYZ is a parallelogram since both pairs of opposite sides of a quadrilateral are congruent. Example 3-1c

Determine whether the quadrilateral is a parallelogram Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Example 3-3a

Determine whether the quadrilateral is a parallelogram Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: One pair of opposite sides is parallel and has the same measure, which means these sides are congruent. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Example 3-3b

Find x so that the quadrilateral is a parallelogram. B C D Answer: When x is 7, ABCD is a parallelogram. Example 3-4a

Find y so that the quadrilateral is a parallelogram. Answer: DEFG is a parallelogram when y is 14. Example 3-4c

Find m and n so that each quadrilateral is a parallelogram. b. Answer: Answer: Example 3-4e

D(1, –1) is a parallelogram. Use the Slope Formula. COORDINATE GEOMETRY Determine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula. Example 3-5a

If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition. Example 3-5b

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula Example 3-5f

Answer: The slopes of and the slopes of Therefore, Answer: The slopes of and the slopes of Therefore, Since opposite sides are parallel, ABCD is a parallelogram. Example 3-5f

Kahoot!

Lesson Summary: Objective: The students will prove quadrilaterals parallelograms.

Preview of the Next Lesson: Objective: The students will be able to prove and apply properties of Rectangles, Rhombuses, and Squares.

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