Download 6-4 Special Parallelograms

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 must be complementary. Because
m ABD = 78, m 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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HELP
GEOMETRY
Special Parallelograms
LESSON 6-4
Additional Examples
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.
5x + 11 = 8x + 2
11 = 3x + 2
9 = 3x
3=x
8x + 2 = 8(3) + 2 = 26
5x + 11 = 5(3) + 11 = 26
Diagonals of a rectangle are congruent.
Subtract 5x from each side.
Subtract 2 from each side.
Divide each side by 3.
Substitute.
The length of each diagonal is 26.
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HELP
GEOMETRY
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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HELP
GEOMETRY
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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HELP
GEOMETRY