Parallelogram
A parallelogram is a quadrilateral with two pairs of opposite, parallel sides. In the figure below are 4 types of parallelograms.
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Sides of a parallelogram
A parallelogram has 4 sides. The opposite sides of a parallelogram are equal in length. In the figure below, side BC is equal to AD in length and side AB is equal to CD in length.
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Diagonal BD cuts parallelogram ABCD into two triangles.. Since AD//BC, ∠ADB = ∠DBC. Since AB//DC, ∠ABD = ∠BDC. According to the Angle-Side-Angle (ASA) postulate, △ABD ≅ △CDB. Therefore, AB ≅ CD and AD ≅ BC.
Interior angles of a parallelogram
A parallelogram has 4 internal angles. The opposite angles of a parallelogram are equal in measure. The consecutive angles are supplementary to each other. The sum of the interior angles is 360°.
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In the diagram above, diagonal BD cuts parallelogram ABCD into two triangles. Since AD//BC, ∠ADB = ∠DBC. Since AB//DC, ∠ABD = ∠BDC. Therefore
∠ABC | = ∠ABD + ∠DBC |
= ∠BDC + ∠ADB | |
= ∠ADC |
Similarly, it can be proven that ∠A = ∠C.
Since ∠ADB = ∠DBC and the sum of the angles of a triangle is 180°,
∠A + ∠B | = ∠A + ∠ABD + ∠DBC |
= ∠A + ∠ABD + ∠ADB | |
= 180° |
Therefore, ∠A and ∠ABC are supplementary.
Similarly, it can be proven that ∠B + ∠C = 180°, ∠C + ∠D = 180°, and ∠A + ∠D = 180°. Also ∠A + ∠B + ∠C + ∠D = 360°.
Diagonals of a parallelogram
A parallelogram has two diagonals. The diagonals of a parallelogram bisect each other. In the figure below diagonals AC and BD bisect each other.
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Special parallelograms
There are three cases when a parallelogram is also another type of quadrilateral.
Rhombus | Rectangle | Square |
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Parallelogram with all sides equal in length | Parallelogram with four right angles | Parallelogram with four right angles and all four sides equal in length |
Symmetry in a parallelogram
A parallelogram has rotational symmetry of order 2.
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Rotating the parallelogram above 180° about point E will result in a parallelogram that is the same as the original shape.
Area of a parallelogram
The area of a parallelogram is the product of the length of its base (b) and height (h).
A = b·h
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In the diagram above, △ABE ≅△DCF. Therefore the area of parallelogram ABCD is equal to the area of rectangle AEFD, or b×h.