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.
---
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 \(x\):
\[
284^\circ + x^\circ = 360^\circ \implies x^\circ = 360^\circ - 284^\circ = 76^\circ.
\]
---
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.
---
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:
\[
\boxed{
\begin{aligned}
1. & \ x^\circ = 26^\circ \\
2. & \ x^\circ = 113^\circ \\
3. & \ x^\circ = 120^\circ \\
4. & \ x^\circ = 114^\circ \\
5. & \ x^\circ = 44^\circ \\
6. & \ x^\circ = 113^\circ \\
7. & \ x^\circ = ?, \ y^\circ = ? \quad \text{(Insufficient information)} \\
8. & \ x^\circ = 122^\circ
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of quadrilateral worksheet algebra.
