Question

two adjacent angles of parallelogram are in the ratio 4:5 find the measure of all the angles​

Answer 1

Step-by-step explanation:

Example 28 : Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures. Solution : Let the angles be 4x and 5x. Then, 4x + 5x = 180° 9x = 180° x = 20° So, angles are 4 × 20° = 80° and 5 × 20° =100°.

Answer 2

★ Concept

The above question is simply based on the properties of Parallelogram. Given that, the two adjacent (situated next to) angle of Parallelogram are in the ratio 4 : 5. Let the common multiple be 'x', thus the adjacent angles (say A & B) would be 4x and 5x respectively. Using one of the basic property i.e, the sum of adjacent angles of Parallelogram is 180°, we will able to calculate the value for 'x' and so does the adjacent angles namely Angle A and B. With due consideration of the figure of Parallelogram Angle C and D must be adjacent to Angle B and A, thus, we will use the same property to find these remaining angles (which are even adjacent to each other).

Let's proceed with Calculation !!

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⋆ Property Used

$\underline{ \boxed{ \red{\sf \: Sum \: of \: adjacent \: angles \: of \: Parallelogram = 180 \degree}}}$

Let the two adjacent angles be ∠A and ∠B.

[ATQ (according to the question), Angle A and B are in the ratio 4 : 5]

Taking Common Multiple be 'x'.

Therefore, ∠A = 4x ; ∠B = 5x

We know that, the sum of adjacent angles of Parallelogram = 180°, we have

$\sf 4x + 5x = 180 \degree$

$\sf9x = 180 \degree \qquad \: x = \dfrac{180}{9} = 20$

$\underline{ \boxed{ \orange{ \bf \: x = 20 \degree}}}$

Measures for Angle A and B -

∠A = 4x = 4(20°) = 80°

∠B = 5x = 5(20°) = 100°

[We cannot assume a Parallelogram (a Quad) having just two angles, right. We are still left with finding the measures for the remaining two angles say Angle C and D adjacent to Angle B and A respectively].

>>∠B + ∠C = 180° (Sum of adjacent angles of Parallelogram)

>> 100° + ∠C = 180°

>> ∠C = 180° - 100° = 80°

Similarly, ∠A + ∠D = 180° (Adjacent angles of Parallelogram)

>> 80° + ∠D = 180°

>> ∠D = 180° - 80° = 100°

Conclusion

Measures of each angles of Parallelogram are 80° , 100°, 80° and 100°. (Angle A, B, C and D respectively).

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Additional Information

A parallelogram is a special type of quadrilateral.

Below are provided some basic properties of Parallelogram.

1) Opposite sides are parallel.

2) Same-Side interior angles (consecutive angles) are supplementary.

3) Each diagonal of a parallelogram separates it into two congruent triangles.

4) The diagonals of a parallelogram bisect each other.