what do all the angles of a parallelogram add up to

Angles of a Parallelogram

There are iv interior angles in a parallelogram and the sum of the interior angles of a parallelogram is e'er 360°. The opposite angles of a parallelogram are equal and the consecutive angles of a parallelogram are supplementary. Let united states of america read more nearly the properties of the angles of a parallelogram in item.

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

Backdrop of Angles of a Parallelogram

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

• The reverse angles of a parallelogram are coinciding (equal). Hither, ∠A = ∠C; ∠D = ∠B.
• All the angles of a parallelogram add upward to 360°. Hither,∠A + ∠B + ∠C + ∠D = 360°.
• All the respective sequent 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 issues 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.

Allow the states learn almost 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 Testify: ∠A =∠C and ∠B=∠D

Proof: In the parallelogram ABCD, diagonal Air-conditioning is dividing the parallelogram into two triangles. On comparison triangles ABC, and ADC. Here we have:
Ac = AC (common sides)
∠1 = ∠four (alternate interior angles)
∠2 = ∠3 (alternate interior angles)
Thus, the ii triangles are congruent, △ABC ≅ △ADC
This gives ∠B = ∠D by CPCT (respective parts of coinciding triangles).
Similarly, we can prove 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, so it is a parallelogram. Let u.s. testify the aforementioned.

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 (Nosotros 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 evidence 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 united states prove this property because the post-obit given fact and using the same figure.

Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively.
To testify: ∠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, information technology is proved that the consecutive angles of a parallelogram are supplementary.

Related Manufactures on Angles of a Parallelogram

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

• Perimeter of Parallelogram
• Parallelogram Worksheets
• Parallelogram Formula
• Properties of Parallelograms

Solved Examples on Angles of a Parallelogram

1. Instance 1: One bending of a parallelogram measures 75°. Find the measure of its adjacent angle and the measure of all the remaining angles of the parallelogram.

   Solution:

   Given that one angle of a parallelogram = 75°
   Let the adjacent angle be x
   Nosotros know that the consecutive (adjacent) angles of a parallelogram are supplementary.
   Therefore, 75° + x° = 180°
   ten = 180° - 75° = 105°
   To find the measure out of all the four angles of a parallelogram we know that the contrary angles of a parallelogram are congruent.
   Hence, ∠i = 75°, ∠ii = 105°, ∠3 = 75°, ∠4 = 105°

2. Example ii: The values of the reverse angles of a parallelogram are given as follows: ∠i = 75°, ∠iii = (10 + thirty)°, find the value of x.
   Given: ∠i and ∠3 are opposite angles of a parallelogram.

   Solution:

   Given: ∠1 = 75° and ∠3 = (x + 30)°
   We know that the opposite angles of a parallelogram are equal.
   Therefore,
   (ten + 30)° = 75°
   x = 75° - thirty°
   x = 45°
   Hence, the value of x is 45°.

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Practice Questions on Angles of a Parallelogram

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FAQs on Angles of a Parallelogram

Practise Angles in a Parallelogram add upwardly to 360°?

Yes, all the interior angles of a parallelogram add together up to 360°. For instance, in a parallelogram ABCD, ∠A + ∠B + ∠C + ∠D = 360°. According to the angle sum holding of polygons, the sum of the interior angles in a polygon tin exist calculated with the assist of the number of triangles that tin be formed within it. In this case, a parallelogram consists of ii triangles, so, the sum of the interior angles is 360°. This can also be calculated by the formula, Southward = (due north − 2) × 180°, where 'n' represents the number of sides in the polygon. Here, 'n' = 4. Therefore, the sum of the interior angles of a parallelogram = Due south = (4 − 2) × 180° = (4 − ii) × 180° = 2 × 180° = 360°.

What is the Relationship Between the Next Angles of a Parallelogram?

The adjacent angles of a parallelogram are besides known as sequent angles and they are e'er supplementary (180°).

How are the Opposite Angles of a Parallelogram Related?

The opposite angles of a parallelogram are e'er equal, whereas, the next angles of a parallelogram are always supplementary.

How to Find the Missing Angles of a Parallelogram?

We can easily discover the missing angles of a parallelogram with the help of three special properties:

• The opposite angles of a parallelogram are congruent.
• The consecutive angles of a parallelogram are supplementary.
• The sum of all the angles of a parallelogram is equal to 360°.

What are the Interior Angles of a Parallelogram?

The angles made on the within of a parallelogram and formed past each pair of side by side sides are its interior angles. The interior angles of a parallelogram sum up to 360° and whatever two side by side (consecutive) angles of a parallelogram are supplementary.

Are all Angles in a Parallelogram Equal?

No, all the angles of a parallelogram are not equal. There are two basic theorems related to the angles of a parallelogram which country that the reverse angles of a parallelogram are equal and the consecutive (adjacent) angles are supplementary.

What is the Sum of the Interior Angles of a Parallelogram?

The sum of the interior angles of a parallelogram is always 360°. Co-ordinate to the bending sum property of polygons, the sum of the interior angles of a polygon can be found by the formula, S = (northward − two) × 180°, where 'n' shows the number of sides in the polygon. In this case, 'n' = 4. Therefore, the sum of the interior angles of a parallelogram = S = (4 − two) × 180° = (4 − 2) × 180° = 2 × 180° = 360°.

Are the Angles of a Parallelogram 90 Degrees?

In some parallelograms like rectangles and squares, all the angles measure out 90°. Yet, the angles in the other parallelograms may not necessarily exist 90°.

Are the Opposite Angles of a Parallelogram Congruent?

Yes, the opposite angles of a parallelogram are congruent. However, the adjacent angles of a parallelogram are always supplementary.

Are Consecutive Angles of a Parallelogram Coinciding?

No, the sequent (next) angles of a parallelogram are not coinciding, they are supplementary.

Are the Contrary Angles of a Parallelogram Supplementary?

No, according to the theorems based on the angles of a parallelogram, the opposite angles are not supplementary, they are equal.

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Source: https://www.cuemath.com/geometry/angles-of-a-parallelogram/

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