Answer: We have proved that when one angle of a parallelogram is 90 0 , the parallelogram is a rectangle. Show that the quadrilateral is a rhombus. First of all, we note that since the diagonals bisect each other, we can conclude that ABCD is a parallelogram. So the opposite sides are equal. Answer: We have proved that the quadrilateral in which the diagonals bisect each other at right angles is a rhombus. The diagonals of ABCD bisect each other at right angles.
Then find the perimeter of ABCD. From Example 2, if the diagonals of a quadrilateral bisect each other at right angles then it becomes a rhombus. There are two important properties of the diagonals of a parallelogram. The diagonal of a parallelogram divides the parallelogram into two congruent triangles. And the diagonals of a parallelogram bisect each other. The diagonals of a parallelogram are NOT equal. The opposite sides and opposite angles of a parallelogram are equal.
A parallelogram is a quadrilateral with opposite sides equal and parallel. The opposite angle of a parallelogram is also equal.
In short, a parallelogram can be considered as a twisted rectangle. It is more of a rectangle, but the angles at the vertices are not right-angles. The square and a rectangle are the two simple examples of a parallelogram. Hence the flat surfaces of the furniture such as a table, a cot, a plain sheet of A4 paper can all be counted as examples of a parallelogram.
The opposite sides of a rectangle are equal and parallel. So a rectangle satisfies all the properties of a parallelogram and hence a rectangle can be called a parallelogram.
Every parallelogram can be called a quadrilateral, but every quadrilateral cannot be called a parallelogram. A trapezium, rhombus, can be called a quadrilateral, but they do not fully satisfy the properties of a parallelogram and hence cannot be called a parallelogram. A square and a rectangle can be called a parallelogram. Learn Practice Download. Properties of Parallelogram Properties of a parallelogram help us to identify a parallelogram from a given set of figures easily and quickly. What are the Properties of Parallelogram?
Properties of Diagonal of Parallelogram 3. Then PQ. This test turns out to be very useful, because it uses only one pair of opposite sides. If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. This test for a parallelogram gives a quick and easy way to construct a parallelogram using a two-sided ruler. Draw a 6 cm interval on each side of the ruler. Joining up the endpoints gives a parallelogram. The test is particularly important in the later theory of vectors.
Then the figure ABQP to the right is a parallelogram. Even a simple vector property like the commutativity of the addition of vectors depends on this construction. The parallelogram ABQP shows, for example, that. This test is the converse of the property that the diagonals of a parallelogram bisect each other.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram:. This test gives a very simple construction of a parallelogram.
Draw two intersecting lines, then draw two circles with different radii centred on their intersection. Join the points where alternate circles cut the lines. This is a parallelogram because the diagonals bisect each other. It also allows yet another method of completing an angle BAD to a parallelogram, as shown in the following exercise. Complete this to a construction of the parallelogram ABCD , justifying your answer. Definition of a Rectangle. A rectangle is a quadrilateral in which all angles are right angles.
Each pair of co-interior angles are supplementary, because two right angles add to a straight angle, so the opposite sides of a rectangle are parallel. This means that a rectangle is a parallelogram, so:. The proof has been set out in full as an example, because the overlapping congruent triangles can be confusing.
The diagonals of a rectangle are equal. Let ABCD be a rectangle. Thus we can draw a single circle with centre M through all four vertices.
If a parallelogram is known to have one right angle, then repeated use of co-interior angles proves that all its angles are right angles. We can construct a rectangle with given side lengths by constructing a parallelogram with a right angle on one corner.
First drop a perpendicular from a point P to a line. We have shown above that the diagonals of a rectangle are equal and bisect each other. Conversely, these two properties taken together constitute a test for a quadrilateral to be a rectangle.
A quadrilateral whose diagonals are equal and bisect each other is a rectangle. As a consequence of this result, the endpoints of any two diameters of a circle form a rectangle, because this quadrilateral has equal diagonals that bisect each other. Thus we can construct a rectangle very simply by drawing any two intersecting lines, then drawing any circle centred at the point of intersection.
The quadrilateral formed by joining the four points where the circle cuts the lines is a rectangle because it has equal diagonals that bisect each other. The remaining special quadrilaterals to be treated by the congruence and angle-chasing methods of this module are rhombuses, kites, squares and trapezia. So let's do that:. We have said and proven that parallelograms have four basic properties: The two pairs of opposite sides are of equal length The two pairs of opposite angles are congruent Two pairs of consecutive angles are supplementary And The diagonals bisect each other We will now show that the converse is true - that if one of these properties holds, the quadrilateral is a parallelogram.
Show that ABCD is a parallelogram Strategy To show that a quadrilateral is a parallelogram using the definition of a parallelogram, we need to show that both pairs of opposite sides are parallel.
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