Angles in a Quadrilateral (Worksheets and Solutions) - Free Printable

To solve for the unknown angle \( x \) in each of the given quadrilaterals, we need to use the fact that the sum of the interior angles of a quadrilateral is always \( 360^\circ \). Let's solve each part step by step.

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Part (a)

The given angles are:
- \( 109^\circ \)
- \( 96^\circ \)
- \( 70^\circ \)
- \( x \)

The sum of the interior angles of a quadrilateral is \( 360^\circ \). Therefore, we can write the equation:
\[
109^\circ + 96^\circ + 70^\circ + x = 360^\circ
\]

First, add the known angles:
\[
109^\circ + 96^\circ + 70^\circ = 275^\circ
\]

Now, substitute this back into the equation:
\[
275^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 275^\circ = 85^\circ
\]

Thus, the value of \( x \) in part (a) is:
\[
\boxed{85^\circ}
\]

---

Part (b)

The given angles are:
- \( 87^\circ \)
- \( 94^\circ \)
- \( 58^\circ \)
- \( x \)

Again, the sum of the interior angles of a quadrilateral is \( 360^\circ \). Therefore, we can write the equation:
\[
87^\circ + 94^\circ + 58^\circ + x = 360^\circ
\]

First, add the known angles:
\[
87^\circ + 94^\circ + 58^\circ = 239^\circ
\]

Now, substitute this back into the equation:
\[
239^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 239^\circ = 121^\circ
\]

Thus, the value of \( x \) in part (b) is:
\[
\boxed{121^\circ}
\]

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Part (c)

The given angles are:
- \( 132^\circ \)
- \( 122^\circ \)
- \( 112^\circ \)
- \( x \)

The sum of the interior angles of a quadrilateral is \( 360^\circ \). Therefore, we can write the equation:
\[
132^\circ + 122^\circ + 112^\circ + x = 360^\circ
\]

First, add the known angles:
\[
132^\circ + 122^\circ + 112^\circ = 366^\circ
\]

Now, substitute this back into the equation:
\[
366^\circ + x = 360^\circ
\]

Solve for \( x \):
\[
x = 360^\circ - 366^\circ = -6^\circ
\]

However, this result is not possible because an angle cannot be negative. Let's recheck the problem statement and ensure all angles are correctly interpreted. If the problem is correct as stated, then there might be an error in the given angles. Assuming the problem is correct, the value of \( x \) would be:
\[
\boxed{-6^\circ}
\]

But typically, such a result indicates a need to re-evaluate the problem setup or constraints.

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Final Answers:

\[
\boxed{85^\circ, 121^\circ, -6^\circ}
\]