A quadrilateral whose two pairs the sides space parallel to each and also the four angles at the vertices room not same to the ideal angle, and also then the square is dubbed a parallelogram. Also, the opposite sides room equal in length.

You are watching: Consecutive angles in a parallelogram are _____.

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Here,

AD = BC (opposite sides)

AB = CD (opposite sides)

Sum of every the 4 angles = 360 degrees

Learn more about the parallel here.

The important properties of angles of a parallel are:

If one edge of a parallelogram is a appropriate angle, then all the angles are appropriate anglesOpposite angle of a parallelogram are equal (or congruent)Consecutive angles room supplementary angle to each other (that way they add up to 180 degrees)
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Opposite angle of a Parallelogram

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In the over parallelogram, A, C and also B, D are a pair of the opposite angles.

Therefore, ∠A = ∠C and also ∠B = ∠D

Also, us have different theorems based on the angles of a parallelogram. They are explained below in addition to proofs.

Opposite angle of a Parallelogram are equal

Theorem: Prove that the opposite angles of a parallelogram room equal.

Given: parallel ABCD.

To prove: ∠B = ∠D and ∠A =∠C

Proof:

In the parallelogram ABCD,

AB || CD and advertisement || BC

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Consider triangle ABC and triangle ADC,

AC = AC (common side)

We understand that alternate interior angles room equal.

∠1 = ∠4

∠2 = ∠3

By ASA congruence criterion, two triangles space congruent to every other.

Therefore, ∠B = ∠D and ∠A =∠C

Hence, it is showed that the opposite angles of a parallelogram are equal.

Consecutive angle of a Parallelogram

Theorem: Prove that any type of consecutive angle of a parallelogram room supplementary.

Given: parallelogram ABCD.

To prove: ∠A + ∠B = 180 degrees, ∠C + ∠D = 180 degrees

Proof:

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AB ∥ CD and ad is transversal.

We know that inner angles ~ above the very same side of a transversal are supplementary.

Therefore, ∠A + ∠D = 180°

Similarly, ∠B + ∠C = 180°, ∠C + ∠D = 180° and also ∠A + ∠B = 180°.

Therefore, the sum of any kind of two adjacent angles that a parallel is equal to 180°.

Hence, that is verified that any kind of two surrounding or consecutive angle of a parallelogram are supplementary.

If one edge is a right angle, climate all 4 angles are ideal angles:

From the over theorem, it deserve to be decided that if one angle of a parallelogram is a right angle (that is same to 90 degrees), then all four angles are right angles. Hence, the will become a rectangle.

Since the surrounding sides are supplementary.

For example, ∠A, ∠B are adjacent angles and ∠A = 90°, then:

∠A + ∠B = 180°

90° + ∠B = 180°

∠B = 180° – 90°

∠B = 90°

Similarly, ∠C = ∠D = 90°

Solved Examples

Example 1:

In the adjoining figure, ∠D = 85° and ∠B = (x + 25)°, uncover the value of x.

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Solution:

Given,

∠D = 85° and also ∠B = (x + 25)°

We understand that opposite angles of a parallelogram space congruent or equal.

Therefore,

(x + 25)° = 85°

x = 85° – 25°

x = 60°

Hence, the value of x is 60.

Example 2: Observe the listed below figure.

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Find the values of x, y and also z.

Solution:

From the offered figure,

y = 112° {since the opposite angle of a parallelogram are equal)

z + 40° + 112° = 180° the sum of consecutive angles is same to 180°

z = 180° – 112° – 40° = 28°

Also, x = 28° {x and x are alternating angles)

Therefore, x = 28°, y = 112° and z = 28°.

Example 3: 

Two adjacent angles of a parallelogram are in the proportion 4 : 5. Find their measures. 

Solution: Given,

The proportion of two adjacent angles that a parallel = 4 : 5

Let 4x and 5x the angles. 

Then, 4x + 5x = 180° the amount of two surrounding angles the a parallelogram is supplementary

9x = 180° 

x = 20° 

Therefore, the angles space 4 × 20° = 80° and also 5 × 20° = 100°.

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Practice Problems

Two adjacent angles the a parallelogram room in the proportion 5 : 1. Uncover all the angles of the parallelogram.The opposite angle of a parallelogram are (3x – 4)° and (2x – 1)°. Find the procedures of all angle of the parallelogram.If one of the inner angles of a parallelogram is 100°, then discover the measure of all the staying angles.