How do you prove that if both pairs of opposite sides of a quadrilateral are equal then it is a parallelogram?

Let us take quadrilateral ABCD, such that AB is equal to CD and BC is equal to AD. Draw a line joining A and C. Since the opposite sides are equal and the corresponding angles are equal. So the quadrilateral ABCD is a parallelogram.

How do you prove a quadrilateral is a parallelogram if the opposite angles are congruent?

THEOREM: If a quadrilateral has 2 sets of opposite sides congruent, then it is a parallelogram. THEOREM: If a quadrilateral has 2 sets of opposite angles congruent, then it is a parallelogram. THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram.

How do you know if opposite sides are parallel?

In order to prove that the opposite sides are parallel, you have to prove that the slope of AB is the same as the slope of CD, and the slope of BC is the same as the slope of AD.

What are opposite sides of a quadrilateral?

Two sides of a quadrilateral are called its opposite sides if they do not have a common endpoint. In the given figure, (AB, DC) and (AD, BC) are two pairs of opposite sides of quadrilateral ABCD.

How many pairs of opposite sides are there in a quadrilateral?

2 pairs
There are 2 pairs of opposite angles in a quadrilateral. The angles connected by the two diagonals are the opposite angles.

What is midpoint theorem maths?

The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

How do you prove a parallelogram has opposite sides Theorem?

Quadrilaterals that are Parallelograms

1. Opposite Sides Theorem Converse: If both pairs of opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.
2. Opposite Angles Theorem Converse: If both pairs of opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.

How do you prove if a quadrilateral is a parallelogram?

Criteria proving a quadrilateral is parallelogram

1. 1) If a quadrilateral has one pair of sides that are both parallel and congruent.
2. 2) If all opposite sides of the quadrilateral are congruent.
3. 3) Both pairs of opposite sides are parallel.
4. 4) Opposite angles are congruent.
5. 5) Diagonals bisect.

What is opposite sides of quadrilateral?

How do you prove that a quadrilateral is a parallelogram?

Show that a quadrilateral is a parallelogram in the coordinate plane. Given a parallelogram, you can use the Parallelogram Opposite Sides Theorem (Theorem 7.3) and the Parallelogram Opposite Angles Theorem (Theorem 7.4) to prove statements about the sides and angles of the parallelogram.

What are the opposite sides of a parallelogram?

Opposite Sides Parallel and Congruent Theorem If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. If BC———ADand BC≅AD—, then ABCD is a parallelogram. Proof Ex. 40, p. 383 A

How do you prove that the lines are parallel in geometry?

Theorem: If a transversal cuts across two lines and the alternate interior angles are congruent, then the lines are parallel The two-column proof proved the quadrilateral is a parallelogram by proving opposite sides were parallel. You can also use the paragraph proof form for any of the six ways.

How do you prove that two triangles are congruent?

They are congruent because they are vertical angles (opposite angles sharing a vertex point). Notice that we have two sides and an angle of both triangles inside the quadrilateral. So, we can use the Side-Angle-Side Congruence Theorem to declare the two triangles congruent.