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.