Angles in Quadrilaterals Worksheets - Math Monks - Free Printable

To solve the problem of finding the missing angles in each quadrilateral, we need to use the property that the sum of the interior angles of any quadrilateral is always 360°. Let's go through each problem step by step.

---

Problem 1:

The quadrilateral is a rectangle. All angles in a rectangle are right angles (90°).

- Given: One angle is labeled as \( x^\circ \).
- Since all angles in a rectangle are 90°:
\[
x = 90
\]

Answer for Problem 1: \( x = 90 \)

---

Problem 2:

The quadrilateral has three known angles: 120°, 95°, and 85°. We need to find the fourth angle \( x \).

- Sum of interior angles of a quadrilateral: \( 360^\circ \)
- Equation:
\[
120 + 95 + 85 + x = 360
\]
- Simplify:
\[
300 + x = 360
\]
\[
x = 360 - 300
\]
\[
x = 60
\]

Answer for Problem 2: \( x = 60 \)

---

Problem 3:

The quadrilateral has three known angles: 120°, 60°, and \( x \). The fourth angle is labeled as \( y \).

- Sum of interior angles:
\[
120 + 60 + x + y = 360
\]
- Simplify:
\[
180 + x + y = 360
\]
\[
x + y = 180
\]

Since the quadrilateral is symmetric and the angles appear to be equal on opposite sides, we can assume \( x = y \). Therefore:
\[
x + x = 180
\]
\[
2x = 180
\]
\[
x = 90
\]
\[
y = 90
\]

Answer for Problem 3: \( x = 90 \), \( y = 90 \)

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Problem 4:

The quadrilateral has three known angles: 68°, \( y \), and \( z \). The fourth angle is labeled as \( x \).

- Sum of interior angles:
\[
68 + y + z + x = 360
\]

However, we do not have enough information to solve for \( x \), \( y \), and \( z \) individually without additional constraints. Assuming the problem intends for us to find one variable, let’s focus on \( x \):

If we assume symmetry or additional properties (e.g., parallel sides), we might infer relationships, but with the given information:
\[
x + y + z = 360 - 68 = 292
\]

Without further details, we cannot uniquely determine \( x \), \( y \), and \( z \). Let’s move to the next problem.

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Problem 5:

The quadrilateral has one known angle: 121°. The other three angles are labeled as \( x \), \( y \), and \( z \).

- Sum of interior angles:
\[
121 + x + y + z = 360
\]
- Simplify:
\[
x + y + z = 360 - 121
\]
\[
x + y + z = 239
\]

Without additional constraints, we cannot uniquely determine \( x \), \( y \), and \( z \). Let’s move to the next problem.

---

Problem 6:

The quadrilateral is a rectangle. One angle is 72°, and the other is labeled as \( x \).

- In a rectangle, opposite angles are equal, and all angles are 90°.
- Since one angle is 72°, the adjacent angle must be:
\[
x = 90
\]

Answer for Problem 6: \( x = 90 \)

---

Problem 7:

The quadrilateral has three known angles: 76°, 104°, and 104°. The fourth angle is labeled as \( 25x + 1 \).

- Sum of interior angles:
\[
76 + 104 + 104 + (25x + 1) = 360
\]
- Simplify:
\[
285 + 25x = 360
\]
\[
25x = 360 - 285
\]
\[
25x = 75
\]
\[
x = \frac{75}{25}
\]
\[
x = 3
\]

Answer for Problem 7: \( x = 3 \)

---

Problem 8:

The quadrilateral has three known angles: 14\( x \) - 7, 93°, and 76°. The fourth angle is labeled as 11\( x \) - 2.

- Sum of interior angles:
\[
(14x - 7) + 93 + 76 + (11x - 2) = 360
\]
- Simplify:
\[
14x - 7 + 93 + 76 + 11x - 2 = 360
\]
\[
25x + 160 = 360
\]
\[
25x = 360 - 160
\]
\[
25x = 200
\]
\[
x = \frac{200}{25}
\]
\[
x = 8
\]

Answer for Problem 8: \( x = 8 \)

---

Problem 9:

The quadrilateral is a rectangle. Two angles are 90°, and one angle is 118°. The fourth angle is labeled as \( x \).

- Sum of interior angles:
\[
90 + 90 + 118 + x = 360
\]
- Simplify:
\[
298 + x = 360
\]
\[
x = 360 - 298
\]
\[
x = 62
\]

Answer for Problem 9: \( x = 62 \)

---

Problem 10:

The quadrilateral has two known angles: 90° and 95°. The other two angles are labeled as \( x \) and \( y \).

- Sum of interior angles:
\[
90 + 95 + x + y = 360
\]
- Simplify:
\[
185 + x + y = 360
\]
\[
x + y = 360 - 185
\]
\[
x + y = 175
\]

Without additional constraints, we cannot uniquely determine \( x \) and \( y \). Let’s move to the next problem.

---

Problem 11:

The quadrilateral has two known angles: 85° and 85°. The other two angles are labeled as \( 10x - 5 \) and \( 8x - 13 \).

- Sum of interior angles:
\[
85 + 85 + (10x - 5) + (8x - 13) = 360
\]
- Simplify:
\[
85 + 85 - 5 - 13 + 10x + 8x = 360
\]
\[
152 + 18x = 360
\]
\[
18x = 360 - 152
\]
\[
18x = 208
\]
\[
x = \frac{208}{18}
\]
\[
x = \frac{104}{9}
\]

Answer for Problem 11: \( x = \frac{104}{9} \)

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x = 90 \\
2. & \ x = 60 \\
3. & \ x = 90, \ y = 90 \\
6. & \ x = 90 \\
7. & \ x = 3 \\
8. & \ x = 8 \\
9. & \ x = 62 \\
11. & \ x = \frac{104}{9}
\end{aligned}
}
\]