Question

Prove that in a parallelogram both pairs of opposite sides are congruent

Answer 1

To prove that in a parallelogram both pairs of opposite sides are congruent, we can use the definition of a parallelogram and the properties of parallel lines.

A parallelogram is defined as a quadrilateral with two pairs of parallel sides. Let's label the vertices of the parallelogram as A, B, C, and D, and let's label the sides as AB, BC, CD, and DA.

Using the definition of a parallelogram, we know that AB is parallel to CD, and BC is parallel to DA.

Now, let's draw a diagonal of the parallelogram, AC, which divides the parallelogram into two congruent triangles, triangle ABC and triangle CDA.

Since triangle ABC and triangle CDA are congruent, we know that their corresponding sides are congruent. In particular, AB is congruent to CD (because they are corresponding sides of the congruent triangles), and BC is congruent to DA (because they are also corresponding sides of the congruent triangles).

Therefore, we have shown that both pairs of opposite sides of a parallelogram are congruent, as required.

Answer 2

To prove that both pairs of opposite sides in a parallelogram are congruent, we need to use the definition of a parallelogram.

A parallelogram is a quadrilateral with two pairs of parallel sides. In other words, opposite sides of a parallelogram are parallel. Let's label the parallelogram as ABCD, with AB and CD as opposite sides, and AD and BC as the other pair of opposite sides.

To prove that AB is congruent to CD, we can use the fact that AB and CD are opposite sides of a parallelogram and therefore parallel. This means that AB and CD have the same length, since if they didn't, the opposite sides would not be parallel.

To prove that AD is congruent to BC, we can use a similar argument. AD and BC are also opposite sides of the parallelogram and therefore parallel. This means that AD and BC have the same length, since if they didn't, the opposite sides would not be parallel.

Therefore, we have proved that in a parallelogram, both pairs of opposite sides are congruent.

example

Sure, let's take the parallelogram with vertices at (0, 0), (4, 2), (6, 5), and (2, 3).

To show that both pairs of opposite sides are congruent, we can use the distance formula to calculate the length of each side.

The length of AB is:

sqrt((4-0)^2 + (2-0)^2) = sqrt(16 + 4) = sqrt(20)

The length of CD is:

sqrt((6-2)^2 + (5-3)^2) = sqrt(16 + 4) = sqrt(20)

So AB and CD have the same length and are congruent.

The length of AD is:

sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13)

The length of BC is:

sqrt((6-4)^2 + (5-2)^2) = sqrt(4 + 9) = sqrt(13)

So AD and BC have the same length and are congruent.

Therefore, we have shown that both pairs of opposite sides in this parallelogram are congruent