Post on 09-Feb-2016
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6.4:Special Parallelograms
Objectives:•To use the properties of diagonals of rhombuses and rectangles•Determine whether a parallelogram is a rhombus or a rectangle
Special Parallelograms
Review of the Properties of a parallelogram:• 2 pairs of opposite parallel sides• 2 pairs of opposite congruent sides• Opposite angles are congruent• Consecutive angles are supplementary• Diagonals bisect each other
Special Parallelograms: Rhombus, Rectangle, Square These figures will inherit ALL the properties above,
AND they will each add their own individual properties
Theorems For a Rhombus:
• Each diagonal of a rhombus bisects 2 angles of the rhombus
• The diagonals of a rhombus are perpendicular bisects &
bisects &
A B
CD DCB BADACBD ADC CBA
BDAC
Examples: Find the measures of the numbered angles in the rhombus.
1
2
3 4
12°
The figure below is a rhombus. Find TQ, TP, and SQ.
Finding Angle Measures• What are the measures of the numbered angles in
rhombus ABCD?
1 90m
2 58m
3 58m
1 3 4 180m m m 90 58 4 180m 148 4 180m
4 32m
Theorem for Rectangle
The diagonals of a rectangle are congruent. (remember, they also are bisected, so all 4 segments created by the intersection are congruent)
ANDBDAC
DEBECEAE
E
EXAMPLE
BD=5y-7 and AC = y + 5. Find the value of y and the length of BE.
E
THEOREMSIf one diagonal of a bisects 2 angles of the
then the is a RHOMBUS.
If the diagonals of a are perpendicular, then the is a RHOMBUS.
If the diagonals of a are congruent, then theis a RECTANGLE.
SQUARE
Remember, a square is a RECTANGLE and a RHOMBUS, so it inherits ALL the properties of a rectangle, rhombus and parallelogram.
Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain your answer.
1. Each diagonal is 15 cm long, and one angle of the quadrilateral has a measure of 45°.
2. The diagonals are congruent, perpendicular, and they bisect each other.