Quadrilaterals worksheet with problems on angle calculations and properties.

Let's solve the problems step by step.

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Problem 1: What is the sum of all interior angles of a quadrilateral?

The sum of the interior angles of any quadrilateral is always:
\[
360^\circ
\]

Answer: \(\boxed{360^\circ}\)

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Problem 2: If four angles of a quadrilateral are in the ratio \(1:2:4:5\). Find the angles.

Let the angles be \(x\), \(2x\), \(4x\), and \(5x\). The sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore:
\[
x + 2x + 4x + 5x = 360^\circ
\]
\[
12x = 360^\circ
\]
\[
x = 30^\circ
\]

Now, calculate each angle:
\[
x = 30^\circ, \quad 2x = 60^\circ, \quad 4x = 120^\circ, \quad 5x = 150^\circ
\]

Answer: \(\boxed{30^\circ, 60^\circ, 120^\circ, 150^\circ}\)

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Problem 3: In a quadrilateral, if two \(\angle s\) are right angles and other two angles are in the ratio \(1:2\). Find these angles.

Let the two right angles be \(90^\circ\) each. Let the other two angles be \(x\) and \(2x\). The sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore:
\[
90^\circ + 90^\circ + x + 2x = 360^\circ
\]
\[
180^\circ + 3x = 360^\circ
\]
\[
3x = 180^\circ
\]
\[
x = 60^\circ
\]

Now, calculate the other angle:
\[
2x = 120^\circ
\]

Answer: \(\boxed{60^\circ, 120^\circ}\)

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Problem 4: In a quadrilateral, if two \(\angle s\) are complementary and other two \(\angle s\) are in the ratio \(4:5\). Find the measure of these two \(\angle s\).

Let the two complementary angles be \(a\) and \(b\). Since they are complementary:
\[
a + b = 90^\circ
\]

Let the other two angles be \(4x\) and \(5x\). The sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore:
\[
a + b + 4x + 5x = 360^\circ
\]
\[
90^\circ + 9x = 360^\circ
\]
\[
9x = 270^\circ
\]
\[
x = 30^\circ
\]

Now, calculate the other two angles:
\[
4x = 4 \times 30^\circ = 120^\circ, \quad 5x = 5 \times 30^\circ = 150^\circ
\]

Answer: \(\boxed{120^\circ, 150^\circ}\)

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Problem 5: Three \(\angle s\) of a quadrilateral are \(49^\circ\), \(70^\circ\), and \(121^\circ\). Find the fourth angle.

Let the fourth angle be \(x\). The sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore:
\[
49^\circ + 70^\circ + 121^\circ + x = 360^\circ
\]
\[
240^\circ + x = 360^\circ
\]
\[
x = 120^\circ
\]

Answer: \(\boxed{120^\circ}\)

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Problem 6: In the given figure, find the value of \(x\).

The figure shows a parallelogram with opposite angles equal and adjacent angles supplementary. The angles are labeled as \(7x\), \(5x\), \(3x\), and \(3x\). Since opposite angles are equal:
\[
7x = 5x \quad \text{(This is not directly useful, so we use the sum of adjacent angles)}
\]

Adjacent angles are supplementary:
\[
7x + 3x = 180^\circ
\]
\[
10x = 180^\circ
\]
\[
x = 18^\circ
\]

Answer: \(\boxed{18}\)

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Problem 7: If four angles of a quadrilateral are in the ratio \(2:3:6:7\). Find the four angles.

Let the angles be \(2x\), \(3x\), \(6x\), and \(7x\). The sum of the interior angles of a quadrilateral is \(360^\circ\). Therefore:
\[
2x + 3x + 6x + 7x = 360^\circ
\]
\[
18x = 360^\circ
\]
\[
x = 20^\circ
\]

Now, calculate each angle:
\[
2x = 40^\circ, \quad 3x = 60^\circ, \quad 6x = 120^\circ, \quad 7x = 140^\circ
\]

Answer: \(\boxed{40^\circ, 60^\circ, 120^\circ, 140^\circ}\)

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Problem 8: In a parallelogram, if one angle is \(90^\circ\), then find the other three angles.

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. If one angle is \(90^\circ\), then the opposite angle is also \(90^\circ\). The other two angles are:
\[
180^\circ - 90^\circ = 90^\circ
\]

Thus, all four angles are:
\[
90^\circ, 90^\circ, 90^\circ, 90^\circ
\]

Answer: \(\boxed{90^\circ, 90^\circ, 90^\circ, 90^\circ}\)

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Problem 9: In the given figure, \(ABCD\) is a parallelogram. Find \(\angle D\).

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Given \(\angle A = 50^\circ\), the opposite angle \(\angle C\) is also \(50^\circ\). The adjacent angle \(\angle B\) is:
\[
180^\circ - 50^\circ = 130^\circ
\]

Since \(\angle D\) is opposite to \(\angle B\):
\[
\angle D = 130^\circ
\]

Answer: \(\boxed{130^\circ}\)

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Problem 10: In the figure \(PQRS\) is a parallelogram. Find \(\angle x\).

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. The given angle at \(R\) is \(35^\circ\), and the angle at \(Q\) is \(95^\circ\). The angle \(x\) is part of the triangle \(PQR\). The sum of the angles in a triangle is \(180^\circ\). Therefore:
\[
x + 35^\circ + 95^\circ = 180^\circ
\]
\[
x + 130^\circ = 180^\circ
\]
\[
x = 50^\circ
\]

Answer: \(\boxed{50^\circ}\)

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Problem 11: In the given figure, \(ABCD\) is a parallelogram. Find \(\angle D\) and \(\angle C\).

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Given \(\angle A = 120^\circ\) and \(\angle B = 60^\circ\), the opposite angles are:
\[
\angle C = \angle A = 120^\circ, \quad \angle D = \angle B = 60^\circ
\]

Answer: \(\boxed{60^\circ, 120^\circ}\)

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Final Answers:

1. \(\boxed{360^\circ}\)
2. \(\boxed{30^\circ, 60^\circ, 120^\circ, 150^\circ}\)
3. \(\boxed{60^\circ, 120^\circ}\)
4. \(\boxed{120^\circ, 150^\circ}\)
5. \(\boxed{120^\circ}\)
6. \(\boxed{18}\)
7. \(\boxed{40^\circ, 60^\circ, 120^\circ, 140^\circ}\)
8. \(\boxed{90^\circ, 90^\circ, 90^\circ, 90^\circ}\)
9. \(\boxed{130^\circ}\)
10. \(\boxed{50^\circ}\)
11. \(\boxed{60^\circ, 120^\circ}\)