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Let me solve both parts of the question step-by-step.

---

### **Part 1: The adjacent angles of a parallelogram are (2x - 4)° and (3x - 1)°. Find the measures of all angles of the parallelogram.**

**Key property of parallelograms:**
The adjacent angles of a parallelogram are supplementary, meaning their sum is 180°.

**Set up the equation:**
\[
(2x - 4) + (3x - 1) = 180
\]

**Simplify:**
\[
2x - 4 + 3x - 1 = 180
\]
\[
5x - 5 = 180
\]
\[
5x = 185
\]
\[
x = 37
\]

**Find the angles:**
- First angle: \((2x - 4) = 2(37) - 4 = 74 - 4 = 70°\)
- Second angle: \((3x - 1) = 3(37) - 1 = 111 - 1 = 110°\)

Since opposite angles of a parallelogram are equal:
- Third angle = First angle = \(70°\)
- Fourth angle = Second angle = \(110°\)

**Final answer for Part 1:**
The angles of the parallelogram are \(70°\), \(110°\), \(70°\), and \(110°\).

---

### **Part 2: The measure of each exterior angle of a regular polygon is 60°. Find the number of sides of the polygon and the measure of each interior angle.**

**Key property of regular polygons:**
The measure of each exterior angle of a regular polygon is given by:
\[
\text{Exterior Angle} = \frac{360°}{n}
\]
where \(n\) is the number of sides.

**Set up the equation:**
\[
60 = \frac{360}{n}
\]

**Solve for \(n\):**
\[
n = \frac{360}{60} = 6
\]

So, the polygon has 6 sides (it is a hexagon).

**Find the measure of each interior angle:**
The sum of an exterior angle and its corresponding interior angle is \(180°\). Therefore:
\[
\text{Interior Angle} = 180° - 60° = 120°
\]

**Final answer for Part 2:**
- The polygon has 6 sides.
- Each interior angle is \(120°\).

---

### **Summary of Answers:**
1. The angles of the parallelogram are \(70°\), \(110°\), \(70°\), and \(110°\).
2. The polygon has 6 sides, and each interior angle is \(120°\).

Let me know if you need further clarifications or details!

---

### **5 Related Questions:**
1. What is the formula for the sum of interior angles of a polygon?
2. How do you find the measure of an exterior angle when the number of sides is known?
3. Can a parallelogram have all four angles equal? If yes, what is it called?
4. What are the properties of opposite angles in a parallelogram?
5. How do you find the measure of each angle in a regular octagon?

### **1 Tip:**
Always remember that the sum of adjacent angles in a parallelogram is \(180°\), and the sum of exterior angles in any polygon is always \(360°\), regardless of the number of sides.

The adjacent angles of a parallelogram are (2x - 4)° and (3x - 1)°. Find the measures of all angles of the parallelogram. The measure of each exterior angle of a regular polygon is 60°. Find the number of sides of the polygon and hence the measure of each interior angle.

Learn how to solve a two-part problem involving the angles of a parallelogram and the number of sides of a regular polygon. Includes key concepts like properties of parallelograms, supplementary angles, and exterior angle formulas. Suitable for Grade 8-10 students.

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