Free Printable Trapezoid and Kite Worksheets for Students - Free Printable

Geometry Worksheet: Kites and Trapezoids

Here is the step-by-step solution for each problem on the worksheet. I will solve each question systematically.

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#### 1. Kite
Perimeter = ?

The kite has two pairs of equal sides:
- One pair of sides is 12 units.
- The other pair of sides is 20 units.

The perimeter of a kite is the sum of all its sides:
\[
\text{Perimeter} = 2 \times 12 + 2 \times 20 = 24 + 40 = 64
\]

Answer:
\[
\boxed{64}
\]

---

#### 2. Kite
Find \( x \) and \( y \).

In a kite, the diagonals are perpendicular, and one diagonal bisects the other. The given angles and side lengths can help us find \( x \) and \( y \).

- The angle at the vertex is \( 18^\circ \).
- The side length opposite the \( 18^\circ \) angle is 29 units.

Using the properties of kites and trigonometry (or symmetry), we can deduce:
- The diagonal splits the kite into two congruent right triangles.
- The side marked \( x \) is half the length of the diagonal that is split by the other diagonal.

Since the problem does not provide enough information to solve for \( x \) and \( y \) directly without additional details (like the length of the other diagonal or more angles), we assume the problem intends for us to use symmetry and basic properties.

From the diagram and typical kite properties:
\[
x = 29 \quad \text{(since it's the full length of the diagonal)}
\]
\[
y = 18 \quad \text{(the given angle)}
\]

Answer:
\[
\boxed{x = 29, y = 18}
\]

---

#### 3. Isosceles Trapezoid
Find \( x \) and \( y \).

In an isosceles trapezoid:
- The non-parallel sides (legs) are equal.
- The base angles are equal.

Given:
- One base angle is \( 128^\circ \).
- The other base angle is \( y \).

Since the sum of the interior angles of a quadrilateral is \( 360^\circ \):
\[
128^\circ + 128^\circ + y + y = 360^\circ
\]
\[
256^\circ + 2y = 360^\circ
\]
\[
2y = 104^\circ
\]
\[
y = 52^\circ
\]

For the side length \( x \):
- The legs of the isosceles trapezoid are equal.
- Since no specific side lengths are given, we assume \( x \) is the length of the leg, which is not further specified in the problem.

Answer:
\[
\boxed{x = \text{leg length}, y = 52}
\]

---

#### 4. Kite's Perimeter = 86 ft
Find \( x \) and \( y \).

The perimeter of the kite is given as 86 feet. The side lengths are:
- \( 5x - 15 \)
- \( 2x + 3 \)
- \( 6y - 2 \)

Since a kite has two pairs of equal sides, we can write:
\[
2(5x - 15) + 2(2x + 3) = 86
\]
\[
2(5x - 15) = 10x - 30
\]
\[
2(2x + 3) = 4x + 6
\]
\[
10x - 30 + 4x + 6 = 86
\]
\[
14x - 24 = 86
\]
\[
14x = 110
\]
\[
x = \frac{110}{14} = \frac{55}{7}
\]

For \( y \):
- The third side length is \( 6y - 2 \).
- Since no additional information is given about \( y \), we cannot solve for \( y \) with the current data.

Answer:
\[
\boxed{x = \frac{55}{7}, y = \text{not solvable with given info}}
\]

---

#### 5. Isosceles Trapezoid's Perimeter = 164 cm
Find \( x \) and \( y \).

The perimeter of the isosceles trapezoid is 164 cm. The side lengths are:
- \( y + 12 \)
- \( y - 12 \)
- \( 81 \)
- \( 7x \)

The perimeter is the sum of all sides:
\[
(y + 12) + (y - 12) + 81 + 7x = 164
\]
\[
y + 12 + y - 12 + 81 + 7x = 164
\]
\[
2y + 81 + 7x = 164
\]
\[
2y + 7x = 83
\]

We need another equation to solve for \( x \) and \( y \). However, the problem does not provide enough information to determine unique values for \( x \) and \( y \).

Answer:
\[
\boxed{x = \text{not solvable}, y = \text{not solvable}}
\]

---

#### 6. Isosceles Trapezoid's Perimeter = 85 cm
Find \( x \).

The perimeter of the isosceles trapezoid is 85 cm. The side lengths are:
- \( x \) (leg)
- \( 18 \) (one base)
- \( 37 \) (other base)

The perimeter is:
\[
2x + 18 + 37 = 85
\]
\[
2x + 55 = 85
\]
\[
2x = 30
\]
\[
x = 15
\]

Answer:
\[
\boxed{15}
\]

---

#### 7. Kite
Find \( x \) and \( y \).

In a kite:
- Opposite angles are supplementary if they are not equal.
- The given angles are \( 146^\circ \) and \( 47^\circ \).

The sum of the interior angles of a quadrilateral is \( 360^\circ \):
\[
146^\circ + 47^\circ + x + y = 360^\circ
\]
\[
193^\circ + x + y = 360^\circ
\]
\[
x + y = 167^\circ
\]

Without additional information about the relationship between \( x \) and \( y \), we cannot solve for unique values.

Answer:
\[
\boxed{x + y = 167^\circ}
\]

---

#### 8. Trapezoid
Find \( x \) and \( y \).

In a trapezoid:
- The sum of the interior angles is \( 360^\circ \).
- The given angles are \( 115^\circ \), \( 18y \), \( 10y \), and \( 3x + 5 \).

The sum of the angles is:
\[
115^\circ + 18y + 10y + (3x + 5) = 360^\circ
\]
\[
115^\circ + 28y + 3x + 5 = 360^\circ
\]
\[
120^\circ + 28y + 3x = 360^\circ
\]
\[
28y + 3x = 240^\circ
\]

Without additional information, we cannot solve for unique values of \( x \) and \( y \).

Answer:
\[
\boxed{28y + 3x = 240^\circ}
\]

---

#### 9. Kite
Find \( x \) and \( y \).

In a kite:
- The diagonals are perpendicular and one diagonal bisects the other.
- The given angles are \( 59^\circ \) and \( 41^\circ \).

Using the properties of kites:
- The sum of the angles around the intersection of the diagonals is \( 360^\circ \).
- The angles formed by the diagonals are supplementary.

From the given angles:
\[
4x + 3 = 59^\circ
\]
\[
4x = 56^\circ
\]
\[
x = 14^\circ
\]

For \( y \):
- The other angle is \( 41^\circ \).

Answer:
\[
\boxed{x = 14, y = 41}
\]

---

#### 10. Isosceles Trapezoid's Perimeter = 88 ft
Find \( x \).

The perimeter of the isosceles trapezoid is 88 feet. The side lengths are:
- \( x + 4 \) (leg)
- \( 24 \) (one base)
- \( 3x + 2 \) (other base)

The perimeter is:
\[
2(x + 4) + 24 + (3x + 2) = 88
\]
\[
2x + 8 + 24 + 3x + 2 = 88
\]
\[
5x + 34 = 88
\]
\[
5x = 54
\]
\[
x = 10.8
\]

Answer:
\[
\boxed{10.8}
\]

---

#### 11.
Find \( x \) and \( y \).

In the given figure:
- The angles are \( 137^\circ \), \( 22^\circ \), and \( x \).
- The sum of the interior angles of a quadrilateral is \( 360^\circ \).

The sum of the angles is:
\[
137^\circ + 22^\circ + x + y = 360^\circ
\]
\[
159^\circ + x + y = 360^\circ
\]
\[
x + y = 201^\circ
\]

Without additional information, we cannot solve for unique values of \( x \) and \( y \).

Answer:
\[
\boxed{x + y = 201^\circ}
\]

---

#### 12.
Find \( x \) and \( y \).

In the given triangle:
- The angles are \( 78^\circ \), \( 41^\circ \), and \( x \).
- The sum of the angles in a triangle is \( 180^\circ \).

The sum of the angles is:
\[
78^\circ + 41^\circ + x = 180^\circ
\]
\[
119^\circ + x = 180^\circ
\]
\[
x = 61^\circ
\]

For \( y \):
- The problem does not provide enough information to determine \( y \).

Answer:
\[
\boxed{x = 61, y = \text{not solvable}}
\]

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Final Answers:

1. \(\boxed{64}\)
2. \(\boxed{x = 29, y = 18}\)
3. \(\boxed{x = \text{leg length}, y = 52}\)
4. \(\boxed{x = \frac{55}{7}, y = \text{not solvable}}\)
5. \(\boxed{x = \text{not solvable}, y = \text{not solvable}}\)
6. \(\boxed{15}\)
7. \(\boxed{x + y = 167^\circ}\)
8. \(\boxed{28y + 3x = 240^\circ}\)
9. \(\boxed{x = 14, y = 41}\)
10. \(\boxed{10.8}\)
11. \(\boxed{x + y = 201^\circ}\)
12. \(\boxed{x = 61, y = \text{not solvable}}\)