Math worksheet for calculating angles in quadrilaterals, featuring Section A and Section B exercises with geometric figures and angle values.

Let's solve the problem step by step.

---

🔷 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 parallel sides are supplementary.
- Rectangle / Right angle: 90°

---

#### ✔ Figure 1: Trapezoid (a)
- Given: One angle = 124°
- Markings show two pairs of equal sides (suggesting an isosceles trapezoid)
- In an isosceles trapezoid, base angles are equal.
- So, the angles on the same side are supplementary (since one pair of sides is parallel).

Let’s assume:
- Top left = 124°
- Then top right = ? → But since it's isosceles, the two top angles should be equal? Wait — actually, in isosceles trapezoids, the base angles are equal, so:
- The two angles adjacent to the longer base are equal.
- The two angles adjacent to the shorter base are equal.

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

Looking at the diagram:
- The figure has one pair of parallel sides (indicated by arrows), and two non-parallel sides marked with ticks (equal length), so it's an isosceles trapezoid.
- The 124° angle is at the top-left. Since it's isosceles, the opposite angle (bottom-right) should also be 124°?
- No — in a trapezoid, the angles on the same leg are supplementary.

Actually:
- Angles adjacent to the same leg are supplementary.
- So, if top-left is 124°, then bottom-left (adjacent along the left leg) must be 180° – 124° = 56°.
- But wait — the unknown is labeled a, which is at the bottom-right corner.

Wait — let's label carefully.

From the diagram:
- Top-left: 124°
- Bottom-right: labeled a
- Sides: top and bottom are parallel (arrows), and the non-parallel sides have tick marks (equal), so isosceles trapezoid.

In an isosceles trapezoid:
- Base angles are equal.
- So, the two angles on the top base are equal.
- The two angles on the bottom base are equal.

But only one angle is given: 124°, at the top-left.

So, top-right must also be 124°.

Then sum of top angles = 124 + 124 = 248°

Total for quadrilateral = 360° → remaining two angles (bottom-left and bottom-right) = 360 – 248 = 112°

Since it's isosceles, these two angles are equal → each = 112 ÷ 2 = 56°

So, a = 56°

✔ a = 56°

---

#### ✔ Figure 2: Parallelogram (b)
- Given: One angle = 73°
- In a parallelogram:
- Opposite angles are equal
- Consecutive angles are supplementary (add to 180°)

So, angle b is opposite to 73° → so b = 73°

Wait — but look at the diagram:
- The 73° is on the bottom-left.
- Angle b is on the bottom-right.
- In a parallelogram, opposite angles are equal, and consecutive angles add to 180°.

So, angle b is adjacent to 73° → so they are consecutive → sum to 180°

So:
b = 180° – 73° = 107°

Wait — but let's double-check the diagram.

The shape has arrows indicating opposite sides are parallel → parallelogram.

Angle at bottom-left = 73°

Then angle at bottom-right (b) is adjacent to it → so they form a straight line across the base → yes, supplementary.

So b = 180° – 73° = 107°

✔ b = 107°

---

#### ✔ Figure 3: Rhombus (d)
- All sides equal (ticks), so it's a rhombus (a type of parallelogram)
- Given: One angle = 115°
- In a rhombus:
- Opposite angles are equal
- Adjacent angles are supplementary

So, the angle d is adjacent to 115° → so d = 180° – 115° = 65°

✔ d = 65°

---

#### ✔ Figure 4: Kite (e)
- Shape has two pairs of adjacent equal sides (ticks)
- Given: Top angle = 83°, bottom angle = 37°
- We need to find e, which is the middle angle (between the two unequal sides)

But wait — in a kite:
- One diagonal is a line of symmetry
- The angles between unequal sides are equal
- But here, the top angle is 83°, bottom is 37°, and e is the angle between the two equal sides?

Wait — looking at the diagram:
- It's a kite with two triangles sharing a vertical axis.
- Top triangle: angle = 83°
- Bottom triangle: angle = 37°
- The angle e is the one at the "waist" — the vertex where the two unequal sides meet.

Wait — actually, from the diagram:
- The kite has two pairs of equal adjacent sides.
- The angle e is the vertex angle between the two equal sides on the left and right.

But the top angle is 83°, bottom is 37°.

But in a kite, the two angles between unequal sides are equal — but here, the angles at the top and bottom are not necessarily equal.

But actually, the diagonal splits the kite into two congruent triangles if it's symmetric.

Wait — better approach: Use total angle sum.

Sum of all angles in a quadrilateral = 360°

Given:
- Top angle = 83°
- Bottom angle = 37°
- The other two angles are equal because of symmetry (kite has one axis of symmetry — the vertical one)

So, let the two side angles be x each.

So:
83 + 37 + x + x = 360
120 + 2x = 360
2x = 240
x = 120

So each of the side angles = 120°

Now, angle e is the angle at the top of the lower triangle — but wait — no.

Wait — in the diagram, e is the angle at the left or right side, between the two equal sides?

Wait — actually, the diagram shows:
- Top point: 83°
- Bottom point: 37°
- Left and right points: both have ticks — meaning those sides are equal in pairs.

But angle e is the angle between the two equal sides on the left and right?

No — look again.

Actually, e is the angle at the top of the lower triangle, but it's shared.

Wait — perhaps e is the angle between the two unequal sides?

Wait — no — the kite has:
- Top triangle: two equal sides from top to left and top to right → so the top angle is 83°
- Bottom triangle: two equal sides from bottom to left and bottom to right → bottom angle is 37°
- The two sides on the left and right are shared.

So the two side angles (left and right) are the ones we need.

But angle e is labeled at the left — between the top and bottom.

Wait — the diagram shows e as the angle at the left vertex.

So:
- At top: 83°
- At bottom: 37°
- At left: e
- At right: ?

But due to symmetry (kite), left and right angles are equal.

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

✔ e = 120°

---

#### ✔ Figure 5: Rectangle-like shape (c)
- Has two right angles (marked with squares)
- One angle = 99°
- Need to find angle c

But wait — the figure has:
- Top-left: angle c
- Top-right: right angle (90°)
- Bottom-right: 99°
- Bottom-left: right angle (90°)

Wait — but sum of angles in quadrilateral = 360°

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

✔ c = 81°

---

✔ Summary of Section A:

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

---

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

#### Trapezoid
- One pair of parallel sides
- Angles on the same side (adjacent to the same leg) are supplementary (add to 180°)
- In isosceles trapezoid: base angles are equal

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

#### Kite
- Two pairs of adjacent sides equal
- One diagonal is a line of symmetry
- Angles between unequal sides are equal (but not necessarily opposite)
- Diagonals are perpendicular
- One pair of opposite angles are equal (the ones between unequal sides)

---

🔷 Section B: Calculate the missing angles

#### ✔ Figure 1: Quadrilateral (f)
- Angles given: 118°, 106°, 60°
- Find f

Sum = 360°

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

✔ f = 76°

---

#### ✔ Figure 2: Irregular shape (g)
- This is not a quadrilateral — it's a pentagon? Wait — no.

Wait — the shape has three angles given: 35°, 53°, 22°, and one reflex angle labeled g.

But it looks like a triangle with a "bite" taken out — actually, this is a quadrilateral with a concave angle.

But let's count the vertices: 4 points → quadrilateral.

But the angle g is marked as a reflex angle (greater than 180°).

But we can still use the sum of interior angles = 360°

But wait — the three angles given are:
- Top: 35°
- Bottom-left: 53°
- Bottom-right: 22°

And g is the fourth angle — but it's drawn as a large angle (concave)

So sum of interior angles = 360°

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

But wait — is this correct?

Yes — even if the angle is reflex, the interior angle at that vertex is still part of the total sum.

So:
g = 360 – 110 = 250°

✔ g = 250°

---

#### ✔ Figure 3: Triangle-like shape (h)
- Has a right angle (square symbol), 176°, and 37°
- Find h

Wait — this is a quadrilateral? But it has four sides?

Wait — look: it has a right angle, 176°, 37°, and h

But sum of angles = 360°

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

✔ h = 57°

Wait — but check: Is this possible?

- 90° (right angle)
- 176° (very close to straight)
- 37°
- h = 57°

Sum: 90 + 176 = 266; 266 + 37 = 293; 293 + 57 = 350 → wait, not 360!

Wait — mistake!

Wait — 90 + 176 = 266
266 + 37 = 303
303 + 57 = 360 → yes, correct.

But wait — 303 + 57 = 360 → yes.

But is h the interior angle?

Yes — the angle labeled h is at the top-left.

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

✔ h = 57°

---

✔ Final Answers:

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

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

---

📌 Observations About Angles in Each Quadrilateral

| Shape | Properties |
|------|-----------|
| Trapezoid | One pair of parallel sides. Angles adjacent to the same leg are supplementary. In isosceles trapezoid, base angles are equal. |
| Parallelogram | Opposite angles are equal. Consecutive angles are supplementary. Opposite sides are parallel and equal. |
| Kite | Two pairs of adjacent sides equal. One diagonal is a line of symmetry. One pair of opposite angles are equal. Diagonals intersect at right angles. |

---

✔ All problems solved!