Geometry worksheet with kite problems to find angle measures and variable values.

Let's solve each problem step by step based on the image you provided. The problems involve kites (a quadrilateral with two pairs of adjacent sides equal), and we need to find:

1. Measures of numbered angles in kites.
2. Values of variables using properties of kites.

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🔷 Key Properties of a Kite:

- Two pairs of adjacent sides are equal.
- One diagonal is a line of symmetry (bisects the other diagonal at right angles).
- The diagonals intersect at 90°.
- The angles between unequal sides are equal.
- The diagonal that connects the vertex angles (the ones between equal sides) bisects the angles.
- The other diagonal is perpendicular to this one and is bisected by it.

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## ✔ Examples 1–3: Find the measure of the numbered angles in each kite

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Example 1:

Given:
- ∠1 = 90°
- ∠2 = 90° – 22 = 68°

But wait — let’s analyze carefully.

We see:
- Triangle ABD is split into two right triangles by diagonal BD.
- Diagonals intersect at right angles → so ∠ABD = 90°
- But ∠1 is labeled as 90°, which is likely the angle at the intersection point (where diagonals meet). So ∠1 = 90° (standard for kite diagonals).

Then:
- In triangle ABC or ABD, we have an angle of 22°.
- Since diagonals are perpendicular, and one angle is 22°, then ∠2 = 90° – 22° = 68°

✔ So:
- ∠1 = 90°
- ∠2 = 68°

> ✔️ Answer:
> ∠1 = 90°, ∠2 = 68°

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Example 2:

Given:
- ∠3 = 100°
- ∠4 = ?

We know:
- In a kite, the diagonal connecting the vertex angles (the ones between equal sides) bisects the angles.
- Also, the sum of angles in any quadrilateral is 360°.

From diagram:
- There is a vertical diagonal splitting the kite into two congruent triangles.
- ∠3 = 100° is the top angle.
- This angle is bisected by the diagonal → so each half is 50°.
- The diagonals intersect at 90°, so the bottom angle is split into two parts.

Let’s label:
- Let’s assume the kite has vertices A, B, C, D.
- Diagonal AC is the axis of symmetry.
- ∠BAC = ∠CAD = 50°
- Diagonals intersect at E → ∠AEB = 90°

Now, look at triangle AEB:
- ∠EAB = 50°
- ∠AEB = 90°
- So ∠ABE = 180° – 50° – 90° = 40°

So ∠4 = 40°

But wait — the box says:
> ∠3 = 100° → ∠4 = (180° – 100°)/2 = 40°

Yes! That matches.

So ∠4 = 40°

> ✔️ Answer:
> ∠3 = 100°, ∠4 = 40°

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Example 3:

Given:
- ∠5 = 90°
- ∠6 = 64° + 20° = 84°?

Wait — let's interpret.

From the diagram:
- Diagonals intersect at 90° → so ∠5 = 90° (angle between diagonals)
- Then, ∠6 is one of the base angles.
- We're told: ∠6 = 64° + 20° = 84°? That seems odd.

Wait — perhaps it's showing a triangle where one angle is 64°, and another is 20°?

Looking at the diagram: there's a triangle formed with angles marked.

Let’s suppose:
- In triangle formed by the diagonal and side, we have angles 64° and 20°?
- But sum of angles in triangle is 180°.

Wait — maybe ∠6 is calculated from a triangle.

Actually, the box says:
> ∠5 = 90°
> ∠6 = 64° + 20° = 84°

That doesn't make sense unless it's saying something else.

Wait — perhaps ∠6 is part of a triangle where one angle is 64°, and another is 20°, but that would be inconsistent.

Alternatively, maybe the kite has a triangle with angles 64° and 20°, and we’re finding ∠6.

But more likely, the expression is misread.

Let me re-express:

Possibility: In a triangle, one angle is 64°, and the other is 20°, so third angle is 96°, but not helpful.

Wait — perhaps it's saying that ∠6 is equal to 64° + 20°?

No — that would be 84°, but why?

Wait — perhaps the diagram shows that one angle is split into 64° and 20°, so total ∠6 = 84°?

Ah! That makes sense.

So if ∠6 is composed of two parts: 64° and 20°, then ∠6 = 84°.

And ∠5 = 90°, which is the right angle at the intersection.

So yes:

> ∠5 = 90°
> ∠6 = 64° + 20° = 84°

✔️ Answer:
∠5 = 90°, ∠6 = 84°

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## ✔ Examples 4–6: Find the value of the variable in each kite

Use properties of kites:
- Diagonals are perpendicular → form right angles.
- One diagonal is bisected by the other.
- Angles between unequal sides are equal.
- Use algebra and angle sums.

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Example 4:

Given:
- x + 2x = 90° (because diagonals intersect at 90°, forming right triangles)
- So: 3x = 90 → x = 30

Then:
- x = 30
- 2x = 60
- y = 60 (since opposite angles or corresponding angles?)
- But also given: 3x = 90 → x = 30 → y = 60

Wait — the box says:
> x + 2x = 90 → 3x = 90 → x = 30
> 3x = 90 → x = 30
> y = 60
> z = 30

So likely:
- In triangle, angles are x, 2x, and 90°
- So x + 2x = 90 → x = 30
- Then 3x = 90 → confirms x = 30
- y = 2x = 60°
- z = x = 30°

✔️ Answer:
- x = 30
- y = 60
- z = 30

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Example 5:

Given:
- 3x + 5 = 4x - 30

Solve:
3x + 5 = 4x - 30
→ 5 + 30 = 4x - 3x
→ 35 = x

So x = 35

Then check:
- Left: 3(35) + 5 = 105 + 5 = 110
- Right: 4(35) - 30 = 140 - 30 = 110 → matches

Also given: ∠7 = 110° → that’s consistent.

Then:
- 3x + 5 = 110 → correct
- 4x - 30 = 110 → correct

So x = 35

✔️ Answer: x = 35

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Example 6:

Two equations:
1. 2(3x + 12) = 3x + 66
2. 2y - 22 = 4y + 16

Let’s solve both.

#### Equation 1:
2(3x + 12) = 3x + 66
→ 6x + 24 = 3x + 66
→ 6x - 3x = 66 - 24
→ 3x = 42
→ x = 14

Check:
Left: 2(3×14 + 12) = 2(42 + 12) = 2(54) = 108
Right: 3×14 + 66 = 42 + 66 = 108 → OK

#### Equation 2:
2y - 22 = 4y + 16
→ -22 - 16 = 4y - 2y
→ -38 = 2y
→ y = -19

But negative angle? Unlikely.

Wait — perhaps signs are wrong.

Let’s double-check:
2y - 22 = 4y + 16
Subtract 2y: -22 = 2y + 16
Subtract 16: -38 = 2y
→ y = -19 → invalid

But maybe the equation is:
2y - 22 = 4y + 16 → no solution? Or typo?

Wait — perhaps the diagram shows angles that must be equal?

Wait — the box says:
> 2y - 22 = 4y + 16
> → y = -19

But negative angle doesn’t make sense.

Wait — perhaps the equation was meant to be:
2y - 22 = 4y - 16? Or different sign?

But as written:
2y - 22 = 4y + 16
→ -22 - 16 = 4y - 2y → -38 = 2y → y = -19

Invalid.

But maybe it’s a trick?

Wait — could it be that the angles are supplementary?

But the problem says "find the value of the variable".

Perhaps the kite has angles set up such that:
- One angle is 2y - 22
- Another is 4y + 16
- And they are equal? Or supplementary?

In a kite, non-vertex angles are equal.

But if they are equal:
2y - 22 = 4y + 16 → same as above → y = -19 → invalid.

If they are supplementary:
(2y - 22) + (4y + 16) = 180
→ 6y - 6 = 180
→ 6y = 186
→ y = 31

Then:
- 2y - 22 = 62 - 22 = 40°
- 4y + 16 = 124 + 16 = 140° → sum = 180 → possible

But in the box, it says:
> 2y - 22 = 4y + 16 → y = -19

That can't be right.

Wait — perhaps the equation is:
2y - 22 = 4y - 16?

Try:
2y - 22 = 4y - 16
→ -22 + 16 = 4y - 2y
→ -6 = 2y → y = -3 → still bad

Or:
2y - 22 = 16 - 4y
→ 2y + 4y = 16 + 22
→ 6y = 38 → y ≈ 6.33 → maybe

But the box says: 2y - 22 = 4y + 16 → y = -19

So either:
- The problem has a typo, or
- The angles are not equal or supplementary, but in a triangle?

Wait — perhaps it's a triangle with angles:
- One angle: 2y - 22
- Another: 4y + 16
- Third: ?

But no third angle given.

Alternatively, maybe the two expressions represent equal angles, but due to labeling error?

Wait — looking back: the box says:
> 2y - 22 = 4y + 16
> → y = -19

But that’s mathematically correct — just leads to negative angle.

But since it’s written, maybe accept it?

But geometrically, angles can’t be negative.

So likely, there is a typo in the problem.

But assuming the equation is as written, we solve it:

2y - 22 = 4y + 16
→ -22 - 16 = 4y - 2y
→ -38 = 2y
→ y = -19

So even though it's invalid, the algebra gives y = -19

But in context, probably a typo.

Wait — perhaps the equation is:
2y - 22 = 4y - 16?

Then:
2y - 22 = 4y - 16
→ -22 + 16 = 4y - 2y
→ -6 = 2y → y = -3 → still bad

Or:
2y - 22 = 16 - 4y
→ 2y + 4y = 16 + 22
→ 6y = 38 → y = 38/6 = 6.333...

Still not nice.

Alternatively, maybe:
2y - 22 = 4y + 16 → but it should be supplementary?

Let’s assume they are supplementary:
(2y - 22) + (4y + 16) = 180
6y - 6 = 180
6y = 186
y = 31

Then:
- 2y - 22 = 62 - 22 = 40°
- 4y + 16 = 124 + 16 = 140° → sum = 180 → valid

And 40° and 140° are possible angles in a kite.

But the problem says: 2y - 22 = 4y + 16 → which implies equality, not supplement.

But if the kite has two angles that are equal, then:
2y - 22 = 4y + 16 → y = -19 → invalid

So unless the diagram shows otherwise, likely typo.

But since the problem states that equation, we must go with it.

So:
> 2y - 22 = 4y + 16
> → y = -19

Even though it’s geometrically impossible, the algebraic solution is y = -19

But wait — maybe the equation is:
2y - 22 = 4y - 16 → still bad

Or:
2y + 22 = 4y + 16 → then 22 - 16 = 4y - 2y → 6 = 2y → y = 3

Then:
2(3) + 22 = 6 + 22 = 28
4(3) + 16 = 12 + 16 = 28 → equal

So maybe it's +22, not -22?

But as written: 2y - 22 = 4y + 16

So unless the diagram has angles labeled differently, we have to accept y = -19.

But in real-world geometry, angles can’t be negative.

Therefore, most likely a typo.

But since the problem presents it as is, we’ll write:

> y = -19

But note: This is likely incorrect due to a sign error in the problem.

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## ✔ Final Answers:

Examples 1–3: Angle Measures

1. ∠1 = 90°, ∠2 = 68°
2. ∠3 = 100°, ∠4 = 40°
3. ∠5 = 90°, ∠6 = 84°

Examples 4–6: Variable Values

4. x = 30, y = 60, z = 30
5. x = 35
6. x = 14, y = -19 → Note: y = -19 is mathematically correct but geometrically invalid; likely a typo in the problem.

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🟩 Summary:

| Example | Answer |
|--------|--------|
| 1 | ∠1 = 90°, ∠2 = 68° |
| 2 | ∠3 = 100°, ∠4 = 40° |
| 3 | ∠5 = 90°, ∠6 = 84° |
| 4 | x = 30, y = 60, z = 30 |
| 5 | x = 35 |
| 6 | x = 14, y = -19 *(possible typo)* |

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Let me know if you'd like a visual explanation or correction of Example 6!