Angles in Quadrilaterals (A) | 4th Grade PDF Geometry Worksheets - Free Printable

Let's solve the problems step by step, using properties of quadrilaterals and the fact that the sum of interior angles in any quadrilateral is always 360°.

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## ✔ SECTION A: Calculate the missing angles

1. First Quadrilateral (Trapezoid? — has one pair of parallel sides shown by tick marks)

- Given angle: 124°
- The two sides marked with single ticks are equal → this suggests it’s an isosceles trapezoid, so base angles are equal.
- But only one angle is given. Let’s use the sum of angles = 360°.

Wait — actually, looking at the diagram:
It’s a quadrilateral with two pairs of adjacent equal sides? No — let’s look again.

Actually, the first shape has:
- One angle = 124°
- Two sides marked with single tick → adjacent sides? Not necessarily indicating equal angles unless it’s a kite or isosceles trapezoid.

But notice: only one angle is labeled, and we’re to find 'a'. However, without more info, we can’t assume symmetry.

Wait — perhaps I misread. Looking again:

Actually, this is likely an isosceles trapezoid, because:
- The top and bottom bases are parallel (indicated by arrow marks on top and bottom sides).
- The non-parallel sides (legs) have single tick marks → meaning they are equal → so it’s an isosceles trapezoid.
- In an isosceles trapezoid, base angles are equal.

So:
- Top left angle = 124° → then top right angle is also 124°? No — wait, in an isosceles trapezoid, each pair of base angles is equal.

Actually, in an isosceles trapezoid:
- Angles adjacent to each leg are equal.
- So if top-left is 124°, then top-right should be equal to bottom-left? No.

Better approach: Use angle sum.

Let me label the vertices clockwise: A (top-left, 124°), B (top-right), C (bottom-right, angle ‘a’), D (bottom-left).

Since top and bottom are parallel (arrow marks), then consecutive angles between the parallel lines are supplementary.

So:
- Angle A + Angle D = 180° → 124° + Angle D = 180° → Angle D = 56°
- Similarly, Angle B + Angle C = 180°

But we don’t know Angle B. However, since legs are equal (tick marks), it’s isosceles → base angles equal.

So:
- Angle A = Angle B = 124°? That can’t be, because then 124 + 124 = 248, plus two more angles would exceed 360.

Wait — correction: In an isosceles trapezoid, the angles on the same base are equal.

Standard rule:
- Each pair of base angles are equal.
- So if top base angles are both x, bottom base angles are both y.
- And x + y = 180° (because of parallel lines).

Given one top angle is 124°, so the other top angle is also 124°? Then bottom angles would be (360 - 2×124)/2 = (360 - 248)/2 = 112/2 = 56°.

Yes! So angle ‘a’ is 56°.

✔ a = 56°

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2. Parallelogram (opposite sides marked with double and single ticks → opposite sides equal)

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

Given angle = 73° → so opposite angle is also 73°.

Then the other two angles (including angle b) are equal and must add up to 360° - 2×73° = 360 - 146 = 214° → so each is 107°.

Since angle b is opposite to the 73° angle? Wait — no.

Looking at the diagram: angle b is adjacent to 73°, so it should be supplementary.

In a parallelogram, adjacent angles are supplementary.

So: b + 73° = 180° → b = 107°

✔ b = 107°

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3. Kite (two pairs of adjacent equal sides — tick marks show two pairs of adjacent sides equal)

In a kite:
- One pair of opposite angles are equal (the ones between unequal sides).
- The other two angles are not necessarily equal.
- But total sum is still 360°.

Given angles: 83° and 37°.

Looking at the diagram: the 83° is at the top, 37° at the bottom point.

The kite has two pairs of equal adjacent sides. Typically, the angles between the unequal sides are equal.

But here, we can just use angle sum.

Let’s denote the four angles:
- Top: 83°
- Bottom: 37°
- Left and right: let’s say e and another angle.

Wait — angle ‘e’ is labeled at the left side.

Actually, in a kite, the axis of symmetry goes through the top and bottom vertices. So the two side angles (left and right) are equal.

So:
- Top = 83°
- Bottom = 37°
- Left = e
- Right = e (symmetric)

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

✔ e = 120°

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4. Trapezoid with right angles (two right angles marked)

This is a right trapezoid.

Given:
- One angle = 99°
- Two angles are 90° (right angles)

Sum of all angles = 360°

So: c + 99° + 90° + 90° = 360°

→ c + 279° = 360° → c = 81°

✔ c = 81°

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5. Rhombus (all sides equal — tick marks on all four sides)

In a rhombus (which is a type of parallelogram):
- Opposite angles are equal.
- Adjacent angles are supplementary.

Given angle = 115° → so opposite angle is also 115°.

Then the other two angles (including d) are equal and sum to 360 - 2×115 = 130° → so each is 65°.

Angle d is adjacent to 115°, so d = 180 - 115 = 65°

✔ d = 65°

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

Trapezoid:

- In an isosceles trapezoid, base angles are equal.
- Consecutive angles between the parallel sides are supplementary (add to 180°).
- In a right trapezoid, two angles are 90°, and the other two add to 180°.

Parallelogram:

- Opposite angles are equal.
- Consecutive angles are supplementary (add to 180°).
- Sum of all angles is 360°.

Kite:

- One pair of opposite angles are equal (usually the ones between the unequal sides).
- The other two angles may be different.
- Sum of all angles is 360°.
- Often symmetric along one diagonal.

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## ✔ SECTION B: Calculate the missing angles

1. Quadrilateral with angles 118°, 106°, 60°, and f

Sum = 360°

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

✔ f = 76°

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2. Concave quadrilateral (arrowhead shape) with angles 35°, 53°, 22°, and g

Even though it’s concave, the sum of interior angles is still 360°.

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

Wait — 250°? That seems large, but in a concave quadrilateral, one interior angle can be reflex (>180°).

Yes, angle g is the “reflex” angle at the inward point.

✔ g = 250°

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3. Quadrilateral with right angle, 176°, 37°, and h

One angle is 90° (marked with square), another is 176°, another is 37°, find h.

Sum = 360°

h = 360 - (90 + 176 + 37) = 360 - 303 = 57°

✔ h = 57°

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## 📝 Final Answers:

Section A:

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

Observations:

- Trapezoid: Base angles equal in isosceles; consecutive angles supplementary.
- Parallelogram: Opposite angles equal; consecutive angles supplementary.
- Kite: One pair of opposite angles equal; symmetric.

Section B:

- f = 76°
- g = 250°
- h = 57°

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✔ All problems solved using angle sum property of quadrilaterals (360°) and specific properties of special quadrilaterals.