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:
\[
109^\circ + 96^\circ + 70^\circ + x = 360^\circ
\]

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

Now, solve for \( x \):
\[
275^\circ + x = 360^\circ
\]
\[
x = 360^\circ - 275^\circ
\]
\[
x = 85^\circ
\]

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

---

Part (b)

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

The sum of the interior angles of a quadrilateral is:
\[
87^\circ + 94^\circ + 58^\circ + x = 360^\circ
\]

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

Now, solve for \( x \):
\[
239^\circ + x = 360^\circ
\]
\[
x = 360^\circ - 239^\circ
\]
\[
x = 121^\circ
\]

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

---

Part (c)

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

The sum of the interior angles of a quadrilateral is:
\[
132^\circ + 122^\circ + 112^\circ + x = 360^\circ
\]

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

Now, solve for \( x \):
\[
366^\circ + x = 360^\circ
\]
\[
x = 360^\circ - 366^\circ
\]
\[
x = -6^\circ
\]

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 since their sum exceeds \( 360^\circ \).

Assuming the problem is correct as stated, the value of \( x \) in part (c) is:
\[
\boxed{-6^\circ}
\]

But typically, such a scenario would indicate a mistake in the problem setup. If you have additional context or corrections, please provide them for further clarification.

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Final Answers:

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