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

Let's solve each of the problems (a), (b), and (c) step by step. These are all quadrilateral angle problems, where we need to find the unknown angle $ x $ using the angle sum property of quadrilaterals.

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🔷 Key Concept:

The sum of the interior angles in any quadrilateral is:
$$
360^\circ
$$

We'll use this to find the missing angle $ x $ in each case.

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a)

Given angles:
- $ 109^\circ $
- $ 96^\circ $
- $ 70^\circ $
- $ x $

Sum of angles = $ 360^\circ $

So,
$$
x = 360^\circ - (109^\circ + 96^\circ + 70^\circ)
$$
$$
x = 360^\circ - 275^\circ = 85^\circ
$$

✔ Answer for a): $ x = 85^\circ $

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b)

Given angles:
- $ 87^\circ $
- $ 94^\circ $
- $ 58^\circ $
- $ x $

Note: One angle is outside, but it's adjacent to an interior angle. Let's analyze.

The $ 58^\circ $ is shown outside, forming a straight line with one of the interior angles. That means the interior angle at that vertex is:
$$
180^\circ - 58^\circ = 122^\circ
$$

Now, the four interior angles are:
- $ 87^\circ $
- $ 94^\circ $
- $ 122^\circ $
- $ x $

Sum:
$$
x = 360^\circ - (87^\circ + 94^\circ + 122^\circ) = 360^\circ - 303^\circ = 57^\circ
$$

✔ Answer for b): $ x = 57^\circ $

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c)

Given angles:
- $ 132^\circ $
- $ 122^\circ $
- $ 112^\circ $ — this is external, so we must find the interior angle.

Again, the $ 112^\circ $ is outside, so the interior angle at that vertex is:
$$
180^\circ - 112^\circ = 68^\circ
$$

Now, the known interior angles are:
- $ 132^\circ $
- $ 122^\circ $
- $ 68^\circ $
- $ x $

Sum:
$$
x = 360^\circ - (132^\circ + 122^\circ + 68^\circ) = 360^\circ - 322^\circ = 38^\circ
$$

✔ Answer for c): $ x = 38^\circ $

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✔ Final Answers:

a) $ \boxed{85^\circ} $
b) $ \boxed{57^\circ} $
c) $ \boxed{38^\circ} $

Let me know if you'd like a diagram explanation or further clarification!