## Angle Sums

## Opening

# Angle Sums

- What do you know about the sum of the angles in this figure (or figures)?

- What do you know about the sum of the angles in this figure (or figures)?

Determine how to find the sum of the angle measures for any polygon.

On a blank sheet of paper, draw a pentagon and a hexagon using the ruler side of your protractor.

- What do you think the sum of the angles for each figure is?
- Measure each angle of the pentagon and hexagon. Find the angle sum for each figure.
- Complete the first four rows of the table based on what you know. Make a prediction for the last row. What do you notice? Do you see a pattern in the way that the angle sums increase?

INTERACTIVE: Angle Sum Table

- Explore the angles of the given polygons by dragging the vertices to change the angles.
- Predict what the sum of the angles will be as you change each polygon. Then select the Show Angle Sum buttons to check your prediction.
- Does the angle sum change as the figure is changed?
- How many acute angles can each figure have? How many obtuse angles can each figure have?

INTERACTIVE: Polygon Angle Sum

Compare the number of sides in the polygon to the angle sum. By how many degrees does the angle sum increase for each additional polygon side?

For each figure in the Polygon Angle Sum interactive, draw lines to help you determine the sum of the angles based on figures you know.

INTERACTIVE: Polygon Angle Sum

How many non-overlapping triangles can you divide each polygon into?

Describe a method for finding the angle sum of any polygon. Support your method with examples from your work.

If a regular polygon has 100 sides, what is the measure of each angle?

- Take notes about your classmates' methods for finding the sum of the angles of any polygon.

As your classmates present, ask questions such as:

- When you added the angle measures in the pentagon and hexagon that you drew, did the measures add up to exactly 540° and 720°? Explain.
- Why does the angle sum of a polygon increase by 180° each time a side is added?
- Can you explain how you determined your method for finding the angle sum in any polygon?
- If the number of sides in a regular polygon keeps increasing, what angle measure are the angles in the polygon getting close to? What shape is the polygon getting close to?

- What is the angle sum for a decagon (10 sides)?
- What is the angle sum for a dodecagon (12 sides)?

Investigate the angle measures for regular polygons. Remember that in a regular polygon, all sides are congruent and all angles are congruent.

- Complete the table. What do you notice?

INTERACTIVE: Polygon Angle Measures

- Write a summary about the sums of angles in polygons.

Check your summary:

- Do you explain how to find the sum of the angles for any polygon?
- Do you explain how you can find the sum of the angles of a polygon by using the sum of the angles of a familiar figure?

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Something I wonder about polygons is …**