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

Let's solve the Angles in Quadrilaterals worksheet step by step.

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🔷 Key Concept:

The sum of interior angles in any quadrilateral is always 360°.
This applies to all types: trapezoids, parallelograms, kites, rhombuses, etc.

We'll use this fact to find missing angles.

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## ✔ Section A: Calculate the missing angles

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🔹 1. First quadrilateral (a)

- Shape: Kite or irregular quadrilateral
- Given: One angle = 124°
- Two sides marked with one tick, two others with one tick → suggests symmetry.
- But we only have one angle.

Wait — look at the markings:
- Two adjacent sides have single ticks, and the opposite side has a double tick? Actually, it looks like:
- The two top-left and bottom-right sides are marked with one tick each.
- The other two sides have two ticks?

Actually, let’s interpret:

It appears to be a kite, with two pairs of adjacent equal sides.

But more importantly, we need to use angle relationships.

However, we only know one angle: 124°.

But wait — look closely: there are two tick marks on two sides, and one tick on the others — maybe indicating symmetry?

But no other angles are given. So perhaps this shape is a parallelogram? No, because only one pair of opposite sides seems parallel (from arrows).

Wait — actually, arrows indicate parallel lines.

Let’s go figure by figure.

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🔸 Figure 1: Find a

- This is a quadrilateral with:
- One angle = 124°
- Sides marked with ticks: two adjacent sides have one tick, two others have one tick — but not symmetric?
- Arrows: No arrows → so no parallel sides indicated.
- But there are tick marks on sides, suggesting some symmetry.

Wait — actually, the tick marks are on sides, not angles.

Looking carefully:

- Top-left and bottom-right sides have one tick
- Top-right and bottom-left have one tick → so all four sides have ticks? Not quite.

Wait: There are three tick marks total?

Let me re-express:

- Top-left side: one tick
- Bottom-right side: one tick
- Top-right side: one tick
- Bottom-left side: one tick

So all sides have one tick — that doesn’t help.

But the arrows show which sides are parallel.

Wait — in this first shape, no arrows are present.

But in the second shape, arrows are used.

So likely, tick marks mean equal-length sides, and arrows mean parallel sides.

So for first shape:
- Only one angle known: 124°
- But no parallel sides shown
- And no other angles given

Wait — that can't be. We must be missing something.

Wait — look again: the shape has two pairs of equal adjacent sides? That would make it a kite.

And in a kite, one diagonal is an axis of symmetry, and angles between unequal sides are equal.

But here, only one angle is given.

But we can't calculate without more info.

Wait — perhaps I'm misreading.

Wait — there is a tick mark on the top-left side and bottom-right side, and another tick on the top-right and bottom-left? Let's count:

- Top-left side: tick
- Top-right side: tick
- Bottom-left side: tick
- Bottom-right side: tick

All four sides have ticks? Then all sides equal? That would be a rhombus?

But no — not necessarily.

Alternatively, maybe opposite sides are equal?

But only one tick per side — hard to tell.

Wait — actually, this shape may be a kite, with two pairs of adjacent equal sides.

But we only know one angle: 124°.

Still not enough.

Wait — unless the ticks are on adjacent sides, meaning two pairs of adjacent equal sides.

But still, only one angle given.

Wait — perhaps the shape is a parallelogram? But no arrows.

Wait — no, the second shape has arrows.

Let’s go to the second shape.

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🔸 Figure 2: Parallelogram (b)

- It has arrows on opposite sides → indicates parallel sides
- One angle = 73°
- In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (add to 180°)

So:
- Angle b is opposite to 73° → so b = 73°

✔ Answer: b = 73°

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🔸 Figure 3: Rhombus or kite? (d)

- All sides marked with ticks → all sides equal → rhombus
- One angle = 115°
- In a rhombus, opposite angles are equal, and adjacent angles are supplementary

So:
- d is adjacent to 115° → so d = 180° − 115° = 65°

✔ Answer: d = 65°

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🔸 Figure 4: Kite (e)

- Two pairs of adjacent equal sides (tick marks)
- Angles: 83° and 37°
- The 83° and 37° are at the ends of the longer diagonal

In a kite, one diagonal is a line of symmetry, and the angles between unequal sides are equal.

But here, we see:
- Two angles given: 83° and 37°
- The unknown angle e is at the "top" vertex

Wait — the shape looks like a kite with:
- Top angle = 83°
- Bottom angle = 37°
- Two other angles are equal (because of symmetry)

Let’s denote the angles:
- Top: 83°
- Bottom: 37°
- Left and right angles: both = e? Wait — no, e is labeled at the left middle?

Wait — the diagram shows:
- Top angle: 83°
- Bottom angle: 37°
- The angle labeled e is at the middle left, between the two triangles

Wait — actually, the shape is divided into two triangles by a diagonal.

So it’s a kite, split along the axis of symmetry.

Then:
- The two upper parts are equal
- The two lower parts are equal

Given:
- Top angle = 83°
- Bottom angle = 37°
- The other two angles (on the sides) are equal due to symmetry

So total sum of angles = 360°

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

✔ Answer: e = 120°

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🔸 Figure 5: Trapezoid (c)

- Right angles at top-right and bottom-left (square corners)
- One angle = 99°
- Need to find angle c (top-left)

This is a trapezoid with two right angles.

Since it has two right angles, and one angle is 99°, and sum is 360°.

Let’s label:
- Top-right: 90°
- Bottom-left: 90°
- Bottom-right: 99°
- Top-left: c = ?

Sum:
- 90 + 90 + 99 + c = 360
- 279 + c = 360
- c = 360 − 279 = 81°

✔ Answer: c = 81°

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🔸 Figure 1: First quadrilateral (a)

Now back to the first one.

Shape: Quadrilateral with:
- One angle = 124°
- Side markings: two sides with one tick, two with one tick — but not clear

Wait — look: the top side has a tick, the right side has a tick, the bottom side has a tick, the left side has a tick — all have ticks?

But more importantly, the arrows — no arrows.

But there are tick marks on sides — possibly indicating equal adjacent sides?

Wait — actually, it might be a kite.

But only one angle given.

Wait — perhaps the 124° angle is at the apex, and the two adjacent sides are equal?

But still, only one angle.

Wait — look again: the bottom-left corner is labeled a, and there are ticks on the two adjacent sides — so those two sides are equal.

Similarly, the top-right side has a tick, and the top-left side has a tick — so maybe two pairs of adjacent equal sides?

So it's a kite.

In a kite:
- One diagonal is axis of symmetry
- The angles between unequal sides are equal
- But we don't know which angles are equal

But we only know one angle: 124°

Is it possible that the 124° angle is between the unequal sides?

Wait — in a kite, the angles between unequal sides are equal, and the other two angles are equal.

But here, only one angle is given.

Wait — unless the 124° angle is one of the equal ones, then we could say the opposite angle is also 124°, and the other two add to 360 − 248 = 112°, so each is 56°?

But we don't know if 124° is paired.

Wait — look at the tick marks:

- Top-left side: tick
- Top-right side: tick
- Bottom-left side: tick
- Bottom-right side: tick

So all four sides have one tick — does that mean all sides are equal?

If all sides are equal, it's a rhombus.

But then it would be a rhombus, and opposite angles equal.

But only one angle is given: 124°

Then opposite angle is 124°, and the other two angles are equal and add to 360 − 248 = 112° → each is 56°

So angle a is at the bottom-left — is it adjacent to 124°?

Yes — so it's not opposite, so it's one of the 56° angles.

But is that correct?

Wait — in a rhombus, adjacent angles are supplementary.

So if one angle is 124°, then adjacent angle is 180 − 124 = 56°

So angle a is adjacent to 124° → so a = 56°

✔ Answer: a = 56°

But wait — is it a rhombus?

Only if all sides are equal.

The tick marks suggest that all sides are equal — yes, each side has a single tick — so likely all sides equal → rhombus.

So yes, a = 56°

✔ Answer: a = 56°

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Summary of Section A:

| Letter | Answer |
|-------|--------|
| a | 56° |
| b | 73° |
| c | 81° |
| d | 65° |
| e | 120° |

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

Trapezoid

- Has one pair of parallel sides
- Consecutive angles between the parallel sides are supplementary (add to 180°)
- If it has two right angles, it's a right trapezoid
- Base angles may be equal if it's isosceles

> Note: In this case, it had two right angles, so the other two added to 180°

Parallelogram

- Opposite angles are equal
- Adjacent angles are supplementary (add to 180°)
- Opposite sides are parallel and equal
- Diagonals bisect each other

Kite

- Two pairs of adjacent sides are equal
- One diagonal is a line of symmetry
- Angles between unequal sides are equal
- Diagonals are perpendicular
- One diagonal is bisected

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## ✔ Section B: Calculate the missing angles

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🔸 Figure 1: Quadrilateral (f)

Angles:
- 118°, 106°, 60°, and f

Sum = 360°

So:
- f = 360 − (118 + 106 + 60) = 360 − 284 = 76°

✔ f = 76°

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🔸 Figure 2: Quadrilateral (g)

This is a concave quadrilateral — shaped like an arrowhead.

Angles:
- 35°, 53°, 22°, and g

Sum = 360°

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

Wait — is that possible?

Yes — because it's a concave quadrilateral, one angle can be reflex (>180°)

So g = 250°

✔ g = 250°

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🔸 Figure 3: Quadrilateral (h)

Angles:
- One angle is 176°
- One is 37°
- One is 90° (right angle)
- h is the missing angle

Sum = 360°

So:
- h = 360 − (176 + 37 + 90) = 360 − 293 = 67°

✔ h = 67°

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## ✔ Final Answers

Section A:

- a = 56°
- b = 73°
- c = 81°
- d = 65°
- e = 120°

Section B:

- f = 76°
- g = 250°
- h = 67°

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## 📌 Observations About Each Quadrilateral:

Trapezoid

- One pair of parallel sides
- Consecutive angles between parallel sides are supplementary
- If isosceles, base angles are equal

Parallelogram

- Opposite sides parallel and equal
- Opposite angles equal
- Adjacent angles supplementary
- Diagonals bisect each other

Kite

- Two pairs of adjacent equal sides
- One diagonal is axis of symmetry
- Angles between unequal sides are equal
- Diagonals intersect at 90°
- One diagonal is bisected

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✔ All problems solved!