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GeometryGeometry

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Find the values of the variables for which GHIJ must be a parallelogram.

1. 2.

x = 6, y = 0.75 a = 34, b = 26

Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram

Lesson 6-3

Determine whether the quadrilateral must be a parallelogram. Explain.

3. 4. 5.

No; both pairs of opposite sides are not necessarily congruent.

Yes; the diagonals bisect each other.

Yes; one pair of opposite sides is both congruent and parallel.

Lesson Quiz

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A segment bisects an angle if and only if it divides the angle into two congruent angles.

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Proof: ABCD is a rhombus, so its sides are all congruent. by the Reflex. POC.AC AC

12 and 34 by CPCTC. Therefore, ABC ADC by the SSS Postulate.

Therefore, bisects BAD and BCD by the definition of bisect.

AC

You can show similarly that bisects ABC and ADC.BD

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In the rhombus above, points B and D are equidistant from A and C. By the Converse of the Perpendicular Bisector Theorem, they are on the perpendicular bisector of segment AC.

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Statements Reasons

1. Rectangle ABCD 1. Given2. Definition of rectangle2. ABCD is a

3. AD BC 3. → opp. sides 4. Definition of rectangle4. DAB & CBA are right s

5. DABCBA 5. All right s are

7. DABCBA 7. SAS8. AC BD

6. AB AB 6. Reflexive POC

8. CPCTC

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GeometryGeometry

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Find the measures of the numbered angles in the rhombus.

Theorem 6-10 states that the diagonals of a rhombus are perpendicular, so m 2 = 90.

Theorem 6–9 states that each diagonal of a rhombus bisects two angles of the rhombus, so m 1 = 78.

Because the four angles formed by the diagonals all must have measure 90, 3 and ABD must be complementary. Because m ABD = 78, m 3 = 90 – 78 = 12.

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Finally, because BC = DC, the Isosceles Triangle Theorem allows you to conclude 1 4. So m 4 = 78.

Quick Check

Additional Examples

6-4

Finding Angle Measures

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GeometryGeometry

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One diagonal of a rectangle has length 8x + 2. The other

diagonal has length 5x + 11. Find the length of each diagonal. By Theorem 6-11, the diagonals of a rectangle are congruent.

The length of each diagonal is 26.

5x + 11 = 8x + 2 Diagonals of a rectangle are congruent.

11 = 3x + 2 Subtract 5x from each side.

9 = 3x Subtract 2 from each side.

3 = x Divide each side by 3.

5x + 11 = 5(3) + 11 = 268x + 2 = 8(3) + 2 = 26 Substitute.

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Quick Check

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Finding Diagonal Length

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GeometryGeometry

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The diagonals of ABCD are perpendicular. AB = 16 cmand BC = 8 cm. Can ABCD be a rhombus or rectangle?Explain.

Use indirect reasoning to show why ABCD cannot be arhombus or rectangle.

Suppose that ABCD is a parallelogram. Then, because its diagonalsare perpendicular, ABCD must be a rhombus by Theorem 6-12.

But AB = 16 cm and BC = 8 cm. This contradicts the requirementthat the sides of a rhombus are congruent. So ABCD cannot be arhombus, or even a parallelogram.

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Quick Check

Additional Examples

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Identifying Special Parallelograms

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GeometryGeometry

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Explain how you could use the properties of diagonals to

stake the vertices of a play area shaped like a rhombus.

One way to stake a play area shaped like a rhombus would be to cut two pieces of rope of any lengths and join them at their midpoints. Then, position the pieces of rope at right angles to each other, and stake their endpoints.

By Theorem 6-7, if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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By Theorem 6-13, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

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Quick Check

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Real-World Connection

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1. The diagonals of a rectangle have lengths 4 + 2x and 6x – 20. Find x

and the length of each diagonal. Find the measures of the numbered angles in each rhombus.

2. 3.

6; each diagonal has length 16.

m 1 = 62, m 2 = 62, m 3 = 56

m 1 = 90, m 2 = 20, m 3 = 20, m 4 = 70

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Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain.

4. Each diagonal is 15 cm long, and one angle of the quadrilateral hasmeasure 45.

5. The diagonals are congruent, perpendicular, and bisect each other.

Yes; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, and if diagonals of a parallelogram are perpendicular, the quadrilateral is a rhombus, and a rectangle that is a rhombus is a square.

Impossible; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, but a rectangle has four right angles.

Lesson Quiz

6-4

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GeometryGeometry

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PACE is a parallelogram and m PAC = 109. Complete each of the following.

1. EC = ? 2. EP = ?

3. m CEP = ? 4. PR = ?

5. RE = ? 6. CP = ?

7. m EPA = ? 8. m ECA = ?

9. Draw a rhombus that is not a square. Draw a rectangle that is not a square. Explain why each is not a square.

(For help, go to Lesson 6-2.)

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Check Skills You’ll Need

Check Skills You’ll Need

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GeometryGeometry

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1. EC = AP = 4.5 2. EP = AC = 7 3. m CEP = m PAC = 109

4. PR = CR = 4.75 5. RE = AR = 3.5 6. CP = 2RC = 2(4.75) = 9.5

7. Since PACE is a parallelogram, AC || PE . By the same-side Interior Angles Theorem, PAC and EPA are supplementary. So, m EPA = 180 – m PAC = 180 – 109 = 71.

Solutions

8. From Exercise 7, m EPA = 71. Since PACE is a parallelogram, opposite angles are congruent. So, m ECA = m EPA = 71.

9. Answers may vary. Samples given.The rhombus is not a square because it has no right s. The rectangle is not a square because all 4 sides aren’t .

Check Skills You’ll Need

6-4