Angles Of A Parallelogram- Theorems, Proofs, Properties - Cuemath

Angles of a Parallelogram

There are four interior angles in a parallelogram and the sum of the interior angles of a parallelogram is always 360°. The opposite angles of a parallelogram are equal and the consecutive angles of a parallelogram are supplementary. Let us read more about the properties of the angles of a parallelogram in detail.

1.	Properties of Angles of a Parallelogram
2.	Theorems Related to Angles of a Parallelogram
3.	FAQs on Angles of a Parallelogram

Properties of Angles of a Parallelogram

A parallelogram is a quadrilateral with equal and parallel opposite sides. There are some special properties of a parallelogram that make it different from the other quadrilaterals. Observe the following parallelogram to relate to its properties given below:

• The opposite angles of a parallelogram are congruent (equal). Here, ∠A = ∠C; ∠D = ∠B.
• All the angles of a parallelogram add up to 360°. Here,∠A + ∠B + ∠C + ∠D = 360°.
• All the respective consecutive angles are supplementary. Here, ∠A + ∠B = 180°; ∠B + ∠C = 180°; ∠C + ∠D = 180°; ∠D + ∠A = 180°

Theorems Related to Angles of a Parallelogram

The theorems related to the angles of a parallelogram are helpful to solve the problems related to a parallelogram. Two of the important theorems are given below:

• The opposite angles of a parallelogram are equal.
• Consecutive angles of a parallelogram are supplementary.

Let us learn about these two special theorems of a parallelogram in detail.

Opposite Angles of a Parallelogram are Equal

Theorem: In a parallelogram, the opposite angles are equal.

Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively.

To Prove: ∠A =∠C and ∠B=∠D

Proof: In the parallelogram ABCD, diagonal AC is dividing the parallelogram into two triangles. On comparing triangles ABC, and ADC. Here we have: AC = AC (common sides) ∠1 = ∠4 (alternate interior angles) ∠2 = ∠3 (alternate interior angles) Thus, the two triangles are congruent, △ABC ≅ △ADC This gives ∠B = ∠D by CPCT (corresponding parts of congruent triangles). Similarly, we can show that ∠A =∠C. Hence proved, that opposite angles in any parallelogram are equal.

The converse of the above theorem says if the opposite angles of a quadrilateral are equal, then it is a parallelogram. Let us prove the same.

Given: ∠A =∠C and ∠B=∠D in the quadrilateral ABCD. To Prove: ABCD is a parallelogram. Proof: The sum of all the four angles of this quadrilateral is equal to 360°. = [∠A + ∠B + ∠C + ∠D = 360º] = 2(∠A + ∠B) = 360º (We can substitute ∠C with ∠A and ∠D with ∠B since it is given that ∠A =∠C and ∠B =∠D) = ∠A + ∠B = 180º . This shows that the consecutive angles are supplementary. Hence, it means that AD || BC. Similarly, we can show that AB || CD. Hence, AD || BC, and AB || CD. Therefore ABCD is a parallelogram.

Consecutive Angles of a Parallelogram are Supplementary

The consecutive angles of a parallelogram are supplementary. Let us prove this property considering the following given fact and using the same figure.

Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively. To prove: ∠A + ∠B = 180°, ∠C + ∠D = 180°. Proof: If AD is considered to be a transversal and AB || CD. According to the property of transversal, we know that the interior angles on the same side of a transversal are supplementary. Therefore, ∠A + ∠D = 180°. Similarly, ∠B + ∠C = 180° ∠C + ∠D = 180° ∠A + ∠B = 180° Therefore, the sum of the respective two adjacent angles of a parallelogram is equal to 180°. Hence, it is proved that the consecutive angles of a parallelogram are supplementary.

Related Articles on Angles of a Parallelogram

Check out the interesting articles given below that are related to the angles of a parallelogram.

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