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 marked as $ 58^\circ $, but it's outside the quadrilateral — actually, it's an exterior angle adjacent to one of the interior angles.

Wait! Let's look carefully.

The $ 58^\circ $ is shown outside, next to a vertex, so it's an exterior angle. 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)

This one has two exterior angles shown: $ 112^\circ $ and $ 122^\circ $. We need to convert them to interior angles.

- The interior angle adjacent to $ 112^\circ $ is:
$$
180^\circ - 112^\circ = 68^\circ
$$
- The interior angle adjacent to $ 122^\circ $ is:
$$
180^\circ - 122^\circ = 58^\circ
$$

Also given: one interior angle is $ 132^\circ $, and we're solving for $ x $.

So the four interior angles are:
- $ 68^\circ $
- $ 132^\circ $
- $ 58^\circ $
- $ x $

Sum:
$$
x = 360^\circ - (68^\circ + 132^\circ + 58^\circ) = 360^\circ - 258^\circ = 102^\circ
$$

✔ Answer for (c): $ x = 102^\circ $

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✔ Final Answers:

a) $ \boxed{85^\circ} $
b) $ \boxed{57^\circ} $
c) $ \boxed{102^\circ} $

Let me know if you'd like diagrams or further explanation!