Calculating Angles in Quadrilaterals Worksheet | Fun and Engaging ... - Free Printable

Let's solve the problem step by step.

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🔷 Section A: Calculate the missing angles

We will use the fact that the sum of interior angles in any quadrilateral is 360°.

Also, we’ll use properties of specific quadrilaterals:

- Parallelogram: Opposite angles are equal; consecutive angles are supplementary (add to 180°).
- Rhombus / Kite: Two pairs of adjacent sides equal; some symmetry.
- Trapezoid: One pair of parallel sides → consecutive angles between the parallel sides are supplementary.
- Rectangle / Square: All angles = 90°.
- Kite: Two pairs of adjacent equal sides; one diagonal bisects the other at right angles.

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#### 🔹 Figure 1: Trapezoid (with marked equal sides and one angle = 124°)

It has two sides marked with a single tick (equal), and one angle is 124°. It’s not a parallelogram since only one pair of sides is marked equal.

But notice: The shape appears to be a kite, because it has two pairs of adjacent equal sides (marked with ticks). Let’s check:

- Two sides on top and bottom have single tick marks — so likely kite or isosceles trapezoid?
- But here, both pairs of adjacent sides are equal? Wait — actually, only two sides are marked as equal (top left and bottom right), and another pair (bottom left and top right) — but no clear pattern.

Wait — let's re-analyze.

Actually, the figure has:
- Top-left and bottom-right sides marked with one tick.
- Bottom-left and top-right sides marked with one tick.

So all four sides are marked with ticks — but they are paired differently.

Wait — it looks like two pairs of adjacent sides equal, which suggests a kite.

But more importantly, look at the angle markings.

We are given:
- One angle = 124°
- We need to find angle a

But wait — there are no other angles given.

But notice: the kite has two pairs of equal adjacent sides, and typically, one pair of opposite angles are equal.

But here, only one angle is given: 124°.

But wait — the diagram shows two sides marked with one tick, and the other two sides also marked with one tick — so it might be a parallelogram?

Wait — if both pairs of opposite sides are equal, then it’s a parallelogram.

But the tick marks: top-left and bottom-right have one tick, and bottom-left and top-right have one tick — this suggests opposite sides are equal, so it's a parallelogram.

Yes! So this is a parallelogram.

In a parallelogram:
- Opposite angles are equal
- Consecutive angles are supplementary (add to 180°)

Given: one angle = 124°

Then the opposite angle is also 124°.

The adjacent angles are:
180° − 124° = 56°

So angle a is 56°

✔ a = 56°

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#### 🔹 Figure 2: Parallelogram with one angle = 73°

Given: one angle = 73°

Opposite angle = 73°

Adjacent angles are supplementary:
180° − 73° = 107°

So angle b = 107°

✔ b = 107°

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#### 🔹 Figure 3: Rhombus (all sides equal) with one angle = 115°

This is a rhombus (all sides marked with ticks).

In a rhombus:
- Opposite angles are equal
- Adjacent angles are supplementary

Given: one angle = 115°

So opposite angle = 115°

Adjacent angle = 180° − 115° = 65°

So angle d = 65°

✔ d = 65°

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#### 🔹 Figure 4: Kite (with two pairs of adjacent equal sides) with angles 83° and 37°

We’re given:
- Top angle = 83°
- Bottom angle = 37°
- Need to find angle e

This is a kite — two pairs of adjacent equal sides.

In a kite:
- One diagonal is an axis of symmetry
- The angles between unequal sides are equal
- But here, we can just use sum of angles

Sum of angles in quadrilateral = 360°

We know:
- One angle = 83°
- One angle = 37°
- But we don’t know the other two yet

Wait — the kite has two equal angles between the equal sides.

But from the diagram:
- The top angle is 83°
- The bottom angle is 37°
- The two side angles are equal (due to symmetry)

Let’s call each of the unknown angles e

So total angles:
83° + 37° + e + e = 360°
120° + 2e = 360°
2e = 240°
e = 120°

✔ e = 120°

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#### 🔹 Figure 5: Right trapezoid (two right angles) with one angle = 99°

We are given:
- One angle = 99°
- Two right angles (90°)
- Need to find angle c

Sum of angles = 360°

So:
90° + 90° + 99° + c = 360°
279° + c = 360°
c = 81°

✔ c = 81°

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✔ Section A Answers:

| Letter | Value |
|--------|-------|
| a | 56° |
| b | 107° |
| c | 81° |
| d | 65° |
| e | 120° |

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📝 Write down anything you notice about the angles in each quadrilateral

#### 🔹 Trapezoid:
- Only one pair of parallel sides
- Angles along the same leg are supplementary (if it's an isosceles trapezoid, base angles are equal)
- In this case, the angles on the same side between the parallel lines add up to 180°

#### 🔹 Parallelogram:
- Opposite angles are equal
- Consecutive angles are supplementary (add to 180°)
- Opposite sides are parallel and equal

#### 🔹 Kite:
- Two pairs of adjacent sides are equal
- One pair of opposite angles are equal (the ones between unequal sides)
- Diagonals intersect at right angles
- One diagonal is a line of symmetry

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🔷 Section B: Calculate the missing angles

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#### 🔹 Figure 1: Irregular quadrilateral with angles: 118°, 106°, 60°, and f

Sum = 360°

So:
118° + 106° + 60° + f = 360°
284° + f = 360°
f = 76°

✔ f = 76°

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#### 🔹 Figure 2: Irregular quadrilateral with angles: 35°, 53°, 22°, and g

Sum = 360°

So:
35° + 53° + 22° + g = 360°
110° + g = 360°
g = 250°

Wait — that seems too large. Is this possible?

Wait — look at the shape. It's a concave quadrilateral — it has a "dent" (reflex angle). That reflex angle is labeled g.

So yes, g could be greater than 180°.

So:
35° + 53° + 22° = 110°
So g = 360° − 110° = 250°

✔ g = 250°

(That makes sense — the reflex angle.)

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#### 🔹 Figure 3: Quadrilateral with a right angle, 37°, 176°, and h

Angles:
- 90° (right angle)
- 37°
- 176°
- h = ?

Sum = 360°

So:
90° + 37° + 176° + h = 360°
303° + h = 360°
h = 57°

✔ h = 57°

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✔ Section B Answers:

| Letter | Value |
|--------|-------|
| f | 76° |
| g | 250° |
| h | 57° |

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✔ Final Answers Summary:

#### Section A
- a = 56°
- b = 107°
- c = 81°
- d = 65°
- e = 120°

#### Section B
- f = 76°
- g = 250°
- h = 57°

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📌 Observations on Angles in Each Quadrilateral:

#### Trapezoid:
- One pair of parallel sides
- Consecutive angles between the parallel sides are supplementary (add to 180°)
- If isosceles, base angles are equal

#### Parallelogram:
- Opposite angles are equal
- Consecutive angles are supplementary
- Opposite sides are parallel and equal

#### Kite:
- Two pairs of adjacent equal sides
- One pair of opposite angles are equal (the ones between unequal sides)
- Diagonals intersect at right angles
- One diagonal is a line of symmetry

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✔ All problems solved!