Lesson: Find missing angles in a quadrilateral | Oak National Academy - Free Printable

Let's solve the problem step by step.

We are given a quadrilateral with some angles labeled, and we need to find the size of angle $ m $.

Step 1: Understand the figure

The shape is a quadrilateral (four-sided polygon), and we are given:

- One angle is $ 107^\circ $
- Another angle is $ 54^\circ $
- A third angle is shown as $ 62^\circ $, but it's outside the shape — this is an external angle, so we need to use it to find the internal angle.

Also, we're asked to find angle $ m $, which is one of the internal angles of the quadrilateral.

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Step 2: Use the external angle to find the internal angle

At the bottom-left corner, there's an external angle of $ 62^\circ $. The internal angle at that vertex is supplementary to this (they form a straight line).

So:
$$
\text{Internal angle} = 180^\circ - 62^\circ = 118^\circ
$$

Now we know three of the four internal angles of the quadrilateral:

- $ 118^\circ $ (from the external 62°)
- $ 107^\circ $
- $ 54^\circ $
- $ m $ (unknown)

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Step 3: Use the sum of interior angles in a quadrilateral

The sum of the interior angles of any quadrilateral is:
$$
(4 - 2) \times 180^\circ = 360^\circ
$$

So:
$$
118^\circ + 107^\circ + 54^\circ + m = 360^\circ
$$

Add the known angles:
$$
118 + 107 = 225 \\
225 + 54 = 279
$$

So:
$$
279^\circ + m = 360^\circ \\
m = 360^\circ - 279^\circ = 81^\circ
$$

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✔ Final Answer:

$$
\boxed{a) \ 81^\circ}
$$

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Explanation Summary:

- We converted the external angle ($62^\circ$) into the internal angle ($180^\circ - 62^\circ = 118^\circ$).
- Summed all known internal angles: $118^\circ + 107^\circ + 54^\circ = 279^\circ$.
- Subtracted from $360^\circ$ to get $m = 81^\circ$.

✔ Correct answer: a) 81°.