Angles in Quadrilaterals Worksheets - Math Monks - Free Printable

To solve the problem of finding the missing angles in each quadrilateral, we use the fact that the sum of the interior angles of any quadrilateral is always 360°. We will apply this principle to each problem step by step.

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Solved Example Recap

The solved example shows:
- Given angles: \(105^\circ\), \(107^\circ\), \(72^\circ\), and an unknown angle \(x^\circ\).
- Sum of interior angles: \(360^\circ\).
- Equation: \(105^\circ + 107^\circ + 72^\circ + x^\circ = 360^\circ\).
- Solving for \(x\):
\[
284^\circ + x^\circ = 360^\circ \implies x^\circ = 360^\circ - 284^\circ = 76^\circ.
\]

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Problem 1

Given angles: \(154^\circ\), \(154^\circ\), \(26^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
154^\circ + 154^\circ + 26^\circ + x^\circ = 360^\circ
\]
\[
334^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 334^\circ = 26^\circ
\]

Answer: \(x^\circ = 26^\circ\)

---

Problem 2

Given angles: \(77^\circ\), \(88^\circ\), \(82^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
77^\circ + 88^\circ + 82^\circ + x^\circ = 360^\circ
\]
\[
247^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 247^\circ = 113^\circ
\]

Answer: \(x^\circ = 113^\circ\)

---

Problem 3

Given angles: \(38^\circ\), \(54^\circ\), \(148^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
38^\circ + 54^\circ + 148^\circ + x^\circ = 360^\circ
\]
\[
240^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 240^\circ = 120^\circ
\]

Answer: \(x^\circ = 120^\circ\)

---

Problem 4

Given angles: \(120^\circ\), \(66^\circ\), \(60^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
120^\circ + 66^\circ + 60^\circ + x^\circ = 360^\circ
\]
\[
246^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 246^\circ = 114^\circ
\]

Answer: \(x^\circ = 114^\circ\)

---

Problem 5

Given angles: \(150^\circ\), \(100^\circ\), \(66^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
150^\circ + 100^\circ + 66^\circ + x^\circ = 360^\circ
\]
\[
316^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 316^\circ = 44^\circ
\]

Answer: \(x^\circ = 44^\circ\)

---

Problem 6

Given angles: \(90^\circ\) (right angle), \(67^\circ\), and two unknown angles \(x^\circ\) and another right angle \(90^\circ\).

#### Solution:
\[
90^\circ + 67^\circ + 90^\circ + x^\circ = 360^\circ
\]
\[
247^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 247^\circ = 113^\circ
\]

Answer: \(x^\circ = 113^\circ\)

---

Problem 7

Given angles: \(348^\circ\) (exterior angle at one vertex), and two unknown angles \(x^\circ\) and \(y^\circ\).

#### Step 1: Find the interior angle corresponding to the exterior angle \(348^\circ\).
The interior angle is:
\[
180^\circ - 348^\circ = -168^\circ \quad \text{(This seems incorrect; let's recheck the problem setup.)}
\]

Assuming the problem meant a different configuration, let's assume the correct interpretation is that the quadrilateral has one angle as \(348^\circ - 180^\circ = 168^\circ\) (interior angle).

#### Step 2: Solve for \(x\) and \(y\).
Given angles: \(168^\circ\), and two unknown angles \(x^\circ\) and \(y^\circ\).

\[
168^\circ + x^\circ + y^\circ + \text{(another angle)} = 360^\circ
\]

Without additional information, we cannot uniquely determine \(x\) and \(y\). Let's assume the problem meant a specific configuration or additional information was provided elsewhere.

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Problem 8

Given angles: \(58^\circ\), \(58^\circ\), \(122^\circ\), and an unknown angle \(x^\circ\).

#### Solution:
\[
58^\circ + 58^\circ + 122^\circ + x^\circ = 360^\circ
\]
\[
238^\circ + x^\circ = 360^\circ
\]
\[
x^\circ = 360^\circ - 238^\circ = 122^\circ
\]

Answer: \(x^\circ = 122^\circ\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x^\circ = 26^\circ \\
2. & \ x^\circ = 113^\circ \\
3. & \ x^\circ = 120^\circ \\
4. & \ x^\circ = 114^\circ \\
5. & \ x^\circ = 44^\circ \\
6. & \ x^\circ = 113^\circ \\
7. & \ x^\circ = ?, \ y^\circ = ? \quad \text{(Insufficient information)} \\
8. & \ x^\circ = 122^\circ
\end{aligned}
}
\]