https://file.aiquickdraw.com/mathbot/file/1728417352844rdy4qizn.jpg

The image shows a "polygon path" consisting of regular polygons with 3, 4, 5, and 6 sides joined together. The goal is to find the measures of the labeled angles (∠1, ∠2, ∠3, ∠4, and ∠5) in different sections of the diagram.

### Step 1: Understand the internal angles of each polygon
- **Triangle (3 sides)**: The sum of the interior angles is \(180^\circ\), so each internal angle is \(\frac{180^\circ}{3} = 60^\circ\).
- **Square (4 sides)**: The sum of the interior angles is \(360^\circ\), so each internal angle is \(\frac{360^\circ}{4} = 90^\circ\).
- **Pentagon (5 sides)**: The sum of the interior angles is \(\left(5-2\right) \times 180^\circ = 540^\circ\), so each internal angle is \(\frac{540^\circ}{5} = 108^\circ\).
- **Hexagon (6 sides)**: The sum of the interior angles is \(\left(6-2\right) \times 180^\circ = 720^\circ\), so each internal angle is \(\frac{720^\circ}{6} = 120^\circ\).

### Step 2: Analyze each question

**a. Find ∠2 and ∠5**:
- ∠2 is part of the triangle, so its internal angle is \(60^\circ\).
- ∠5 is part of the pentagon, so its internal angle is \(108^\circ\).

Thus:
- \(m\angle 2 = 60^\circ\)
- \(m\angle 5 = 108^\circ\)

**b. Find ∠3 and ∠4**:
- ∠3 is part of the square, so its internal angle is \(90^\circ\).
- ∠4 is part of the pentagon, so its internal angle is \(108^\circ\).

Thus:
- \(m\angle 3 = 90^\circ\)
- \(m\angle 4 = 108^\circ\)

**c. Find ∠1**:
- ∠1 is part of the hexagon, so its internal angle is \(120^\circ\).

Thus:
- \(m\angle 1 = 120^\circ\)

### Final Answers:
a. 
- \(m\angle 2 = 60^\circ\)
- \(m\angle 5 = 108^\circ\)

b.
- \(m\angle 3 = 90^\circ\)
- \(m\angle 4 = 108^\circ\)

c.
- \(m\angle 1 = 120^\circ\)

---

Would you like more details on how the internal angles of polygons are calculated or have any other questions?

Here are 5 follow-up questions:
1. How are the sum of interior angles of a polygon calculated in general?
2. Why does a regular polygon have equal internal angles?
3. How can you use external angles to verify the correctness of internal angles?
4. Can this method be applied to irregular polygons?
5. How would the calculation change for polygons with more than 6 sides?

**Tip**: The sum of the interior angles of any polygon is given by \((n-2) \times 180^\circ\), where \(n\) is the number of sides.

In Ms. Rickets’ math class, students made a 'polygon path' that consists of regular polygons of 3, 4, 5, and 6 sides joined together. Find the measures of angles ∠1, ∠2, ∠3, ∠4, and ∠5.

Learn how to calculate the interior angles in a 'polygon path' consisting of polygons with 3, 4, 5, and 6 sides. Find the angle measures of ∠1, ∠2, ∠3, ∠4, and ∠5 using geometry concepts like the interior angle sum formula. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation