How to find angles in a parallelogram with diagonals

One special kind of polygons is called a parallelogram. It is a quadrilateral where both pairs of opposite sides are parallel.

There are six important properties of parallelograms to know:

1. Opposite sides are congruent (AB = DC).
2. Opposite angels are congruent (D = B).
3. Consecutive angles are supplementary (A + D = 180°).
4. If one angle is right, then all angles are right.
5. The diagonals of a parallelogram bisect each other.
6. Each diagonal of a parallelogram separates it into two congruent triangles.
   

$$\triangle ACD\cong \triangle ABC$$

If we have a parallelogram where all sides are congruent then we have what is called a rhombus. The properties of parallelograms can be applied on rhombi.

If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid. The parallel sides are called bases while the nonparallel sides are called legs. If the legs are congruent we have what is called an isosceles trapezoid.

In an isosceles trapezoid the diagonals are always congruent. The median of a trapezoid is parallel to the bases and is one-half of the sum of measures of the bases.

$$EF=\frac{1}{2}(AD+BC)$$

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Find the length of EF in the parallelogram

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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.

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)

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.

1. Example 1: One angle 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 We know that the consecutive (adjacent) angles of a parallelogram are supplementary. Therefore, 75° + x° = 180°

   x = 180° - 75° = 105°

   To find the measure of all the four angles of a parallelogram we know that the opposite angles of a parallelogram are congruent.

   Hence, ∠1 = 75°, ∠2 = 105°, ∠3 = 75°, ∠4 = 105°

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FAQs on Angles of a Parallelogram

Yes, all the interior angles of a parallelogram add up to 360°. For example, in a parallelogram ABCD, ∠A + ∠B + ∠C + ∠D = 360°. According to the angle sum property of polygons, the sum of the interior angles in a polygon can be calculated with the help of the number of triangles that can be formed inside it. In this case, a parallelogram consists of 2 triangles, so, the sum of the interior angles is 360°. This can also be calculated by the formula, S = (n − 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 = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° = 360°.

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

The adjacent angles of a parallelogram are also known as consecutive angles and they are always supplementary (180°).

How are the Opposite Angles of a Parallelogram Related?

The opposite angles of a parallelogram are always equal, whereas, the adjacent angles of a parallelogram are always supplementary.

How to Find the Missing Angles of a Parallelogram?

We can easily find 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 inside of a parallelogram and formed by each pair of adjacent sides are its interior angles. The interior angles of a parallelogram sum up to 360° and any two adjacent (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 state that the opposite 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°. According to the angle sum property of polygons, the sum of the interior angles of a polygon can be found by the formula, S = (n − 2) × 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 − 2) × 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 90°. However, the angles in the other parallelograms may not necessarily be 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 Congruent?

No, the consecutive (adjacent) angles of a parallelogram are not congruent, they are supplementary.

Are the Opposite 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.