Special Parallelograms LESSON 6-4 Additional Examples

Find the measures of the numbered angles in the rhombus.

Theorem 6 –9 states that

each diagonal of a rhombus bisects two angles of the rhombus,

so

m

1 = 78.

Theorem 6-10 states that

the diagonals of a rhombus are perpendicular,

so

m

2 = 90.

Because the four angles formed by the diagonals all must have measure 90, 3 and

ABD m ABD

= 78,

m

must be complementary. Because 3 = 90 – 78 = 12.

Finally, because

BC

=

DC

, the Isosceles Triangle Theorem allows you to conclude 1 4. So

m

4 = 78.

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Special Parallelograms LESSON 6-4 Additional Examples

One diagonal of a rectangle has length 8

x

+ 2. The other diagonal has length 5

x

+ 11. Find the length of each diagonal.

By Theorem 6-11, the diagonals of a rectangle are congruent. 5

x

+ 11 = 8

x

+ 2 11 = 3

x

+ 2 9 = 3

x

3 =

x

8

x

+ 2 = 8( 3 ) + 2 = 26 5

x

+ 11 = 5( 3 ) + 11 = 26 The length of each diagonal is 26.

Diagonals of a rectangle are congruent.

Subtract 5

x

from each side.

Subtract 2 from each side.

Divide each side by 3.

Substitute.

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Special Parallelograms LESSON 6-4 Additional Examples

The diagonals of

ABCD

are perpendicular.

AB

= 16 cm and

BC

= 8 cm. Can

ABCD

be a rhombus or rectangle?

Explain.

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

Suppose that

ABCD

is a parallelogram. Then, because its diagonals are perpendicular,

ABCD

must be a rhombus by Theorem 6-12.

But

AB

= 16 cm and

BC

= 8 cm. This contradicts the requirement that the sides of a rhombus are congruent. So

ABCD

cannot be a rhombus, or even a parallelogram.

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Special Parallelograms LESSON 6-4 Additional Examples

Explain how you could use the properties of diagonals to stake the vertices of a play area shaped like a rhombus.

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

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

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