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Solve the understanding quadrilaterals worksheet

I'll review the worksheet and extract the problems on understanding quadrilaterals so I can solve them for you. Allow me a moment.Here are the questions I found related to understanding quadrilaterals, along with solutions:

### 1. **Find the missing values**
This refers to a polygon where certain interior/exterior angle measures are missing. To solve such questions, we use the formula for the sum of interior angles of a polygon:
\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\]
where \( n \) is the number of sides.

For the exterior angles, each angle is:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]

Apply these formulas based on the specific number of sides provided in the worksheet.

### 2. **In a parallelogram ABCD, find the values of \(x\), \(y\), and \(z\)**  
Parallelograms have properties where opposite angles are equal, and consecutive angles are supplementary (i.e., their sum is \(180^\circ\)). Use these properties to solve for the unknowns.

### 3. **What will be the exterior angle of a polygon if each of its interior angles is \(155^\circ\)?**  
The exterior angle of any polygon is calculated using:
\[
\text{Exterior angle} = 180^\circ - \text{Interior angle}
\]
For this problem:
\[
\text{Exterior angle} = 180^\circ - 155^\circ = 25^\circ
\]

### 4. **Two adjacent angles of a parallelogram are in the ratio 4:3. What will be the angles?**
For a parallelogram, the sum of adjacent angles is \(180^\circ\). Let the angles be \(4x\) and \(3x\), then:
\[
4x + 3x = 180^\circ
\]
\[
7x = 180^\circ \quad \Rightarrow \quad x = \frac{180^\circ}{7} = 25.71^\circ
\]
Thus, the angles are \(4x = 102.86^\circ\) and \(3x = 77.14^\circ\).

### 5. **Find each exterior angle of a regular pentagon**
For any regular polygon:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]
For a pentagon (\(n = 5\)):
\[
\text{Exterior angle} = \frac{360^\circ}{5} = 72^\circ
\]

Would you like a detailed explanation for any of these solutions or help with other questions from the worksheet?

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Here are five related questions to deepen your understanding:
1. How do you find the interior angle of a regular hexagon?
2. What are the properties of the diagonals in a rhombus?
3. How do you determine the sum of the exterior angles for any polygon?
4. What conditions must a quadrilateral meet to be classified as a rectangle?
5. How do you calculate the length of diagonals in a square given its side length?

**Tip:** Always remember that the sum of the exterior angles of any polygon, regardless of its number of sides, is always \(360^\circ\).

Solve problems related to understanding quadrilaterals, including finding missing angles in polygons, properties of parallelograms, and calculating exterior angles of regular polygons. This worksheet covers key Grade 8 geometry concepts and provides step-by-step solutions.

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