Angles in a Quadrilateral - Free Printable

Let's solve each part of the problem step by step. The learning objective is to calculate missing angles in a quadrilateral.

🔷 Key Concept:

The sum of the interior angles in any quadrilateral is always:

> 360°

We'll use this fact to find the unknown angles in each case.

---

a)

Given angles:
- 45°
- 124°
- 135°
- Unknown angle = a

Sum of angles = 360°

So:
$$
a = 360° - (45° + 124° + 135°)
$$
$$
a = 360° - 304° = 56°
$$

✔ Answer: a = 56°

---

b)

Given angles:
- Right angle = 90° (marked with a square)
- 70°
- 124°
- Unknown angle = b

Sum = 360°

$$
b = 360° - (90° + 70° + 124°)
$$
$$
b = 360° - 284° = 76°
$$

✔ Answer: b = 76°

---

c)

Given angles:
- One angle = 288° (this is an exterior angle, shown outside the shape)
- Two right angles = 90° each (marked with squares)

Wait — the 288° is outside the shape and appears to be an exterior angle, but we need to be careful.

But looking closely: it's marked outside the quadrilateral at one vertex, so it's likely the exterior angle at that corner.

But we can’t directly use exterior angles unless we know which angle it corresponds to.

However, let’s analyze the diagram:

- Two angles are right angles (90°), marked with squares.
- One angle has an exterior angle of 288°.
- We are to find angle c, which is the interior angle at the top-right corner.

But if the exterior angle is 288°, then the interior angle at that vertex would be:

$$
\text{Interior angle} = 360° - 288° = 72°
$$

Wait! That seems off — because the sum of angles in a quadrilateral is only 360°, and if two angles are 90°, and one is 72°, then:

Let’s suppose:
- Angle A = 90° (bottom-left)
- Angle B = 90° (bottom-right)
- Angle C = ? (top-right) → we're told the exterior angle here is 288°, so interior = 360° - 288° = 72°
- Then angle D = c (top-left)

Wait — but now total interior angles:
$$
90° + 90° + 72° + c = 360°
\Rightarrow 252° + c = 360°
\Rightarrow c = 108°
$$

But wait — the exterior angle of 288° implies that the interior angle is 72°, and that’s fine.

But the question asks for angle c, which is labeled at the top-right vertex.

Looking at the diagram again:

- The 288° is the exterior angle at the top-left vertex (blue arc).
- The c is the interior angle at the top-right vertex.

Wait — actually, the blue arc with 288° is outside the quadrilateral, and it's adjacent to the top-left corner.

So:
- Exterior angle at top-left = 288° → interior angle = 360° − 288° = 72°

Now, we have:
- Top-left interior angle = 72°
- Bottom-left = 90° (right angle)
- Bottom-right = 90° (right angle)
- Top-right = c (unknown)

So:
$$
72° + 90° + 90° + c = 360°
\Rightarrow 252° + c = 360°
\Rightarrow c = 108°
$$

✔ Answer: c = 108°

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d)

This quadrilateral looks like a kite (two pairs of adjacent equal sides, indicated by tick marks).

Given:
- One angle = 127° (top)
- One angle = 67° (bottom)
- Two angles marked with red arcs and labeled d — they are equal (since it's symmetric and marked with same arcs)

Also, the kite has symmetry: two angles on the "sides" are equal.

Let’s label:
- Top angle = 127°
- Bottom angle = 67°
- Left and right angles = d each (since marked with same arcs and symmetrical)

Sum of angles = 360°

So:
$$
127° + 67° + d + d = 360°
\Rightarrow 194° + 2d = 360°
\Rightarrow 2d = 166°
\Rightarrow d = 83°
$$

✔ Answer: d = 83°

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✔ Final Answers:

- a) $ \boxed{56^\circ} $
- b) $ \boxed{76^\circ} $
- c) $ \boxed{108^\circ} $
- d) $ \boxed{83^\circ} $

Let me know if you'd like a visual explanation or further breakdown!