# 4 Sided Concave Polygon Homework

**Hint 1**: If an interior angle is acute, the exterior angle is obtuse, and vice versa.

**Hint 2**: What do the exterior angles of a convex polygon sum to?

**Added later**: The answer to the OP's question can be $6$, $7$, or $\infty$, depending on the precise definitions of "convex" and "obtuse."

If the interior angles of a convex polygon are required to be *strictly* less than $180$ degrees, then the answer is $6$, as Robert Z's answer shows.

If interior angles of $180$ degrees are allowed, and these are considered obtuse (because $180\gt90$), then the answer is $7$: Take, for example, a square, with $4$ right angles and designate any $3$ points along any of the sides of the square as additional vertices, each with an obtuse angle of $180$ degrees. By Robert Z's argument (and/or the hints above), you can't do better than $7$.

Finally, if interior angles of $180$ degrees are allowed, but obtuse angles are required to be *strictly* between $90$ and $180$ degrees (which seems to be a standard definition of "obtuse"), then you can take any convex polygon with three obtuse angles and designate as many more points as you please along the perimeter as additional vertices, each with a (non-obtuse!) interior angle of $180$ degrees. In this case, there is no upper bound on the number of sides, so we can say the answer is $\infty$.

In comments, the OP indicated that convexity does allow for interior angles of $180$ degrees but did not specify whether they are considered obtuse. In either case, the answer $6$ is technically wrong for this definition.

## Presentation on theme: "Polygon Worksheet 1. Concave Polygon Convex Polygon."— Presentation transcript:

1 Polygon Worksheet1.Concave PolygonConvex Polygon

2 **Classifying Polygons 2. Naming Polygons based on the number of sides:**

Triangle (This is labled based on the number of angles.)QuadrilateralPentagonHexagonHeptagonOctagonNonagonDecagon

3 **Triangle - A polygon with three sides.**

PolygonsTriangle - A polygon with three sides.

4 **Quadrilateral - A polygon with four sides.**

PolygonsQuadrilateral - A polygon with four sides.

5 **Parallelogram - A quadrilateral with two pairs of parallel sides.**

PolygonsParallelogram - A quadrilateral with two pairs of parallel sides.

6 **Trapezoid - A quadrilateral which has one pair of parallel sides.**

PolygonsTrapezoid - A quadrilateral which has one pair of parallel sides.

7 **Rectangle - a quadrilateral with four congruent angles (all 90°).**

PolygonsRectangle - a quadrilateral with four congruent angles (all 90°).

8 **Rhombus - A parallelogram with four congruent sides.**

PolygonsRhombus - A parallelogram with four congruent sides.

9 PolygonsSquare - a quadrilateral with four congruent sides and four congruent angles (all 90°).

10 **Pentagon - A polygon with five sides.**

PolygonsPentagon - A polygon with five sides.

11 **Hexagon - A polygon with six sides.**

PolygonsHexagon - A polygon with six sides.

12 **Octagon - A polygon with eight sides.**

PolygonsOctagon - A polygon with eight sides.

13 **Decagon - A polygon with ten sides.**

PolygonsDecagon - A polygon with ten sides.

14 **Polygon Worksheet Sum of Interior Angles**

3. How do you find the sum of the interior angles of an n-gon?Sum of Interior Angles

15 **3. Find the sum of the interior angles of the following polygons.**

Polygon Worksheet3. Find the sum of the interior angles of the following polygons.Hexagon: gonSum of Interior AnglesHexagonSum of Interior Angles22-gon

16 **3. What does “sum of the interior angles” mean?**

Polygon Worksheet3. What does “sum of the interior angles” mean?The answer when I add up all of the angles on the INSIDE of a polygon.

17 **4. What is a regular polygon?**

Polygon Worksheet4. What is a regular polygon?All of the angles are the congruent (same measure) and all of the sides are congruent (same length).

18 **Measure of One Interior angle of a Regular n-gon**

Polygon Worksheet3. How do you find the measure of an (each/one) interior angle of a regular n-gon?Measure of One Interior angle of a Regular n-gon

19 **Polygon Worksheet Measure of one Interior Angle**

3. Find the measure of each interior angle of the following polygons.Hexagon: NonagonHexagonNonagonMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior AngleMeasure of one Interior Angle

20 **4. What is the sum of the exterior angles of any regular polygon?**

Polygon Worksheet4. What is the sum of the exterior angles of any regular polygon?The sum of the measures of the exterior angles of any polygon is always360

21 **Measure of One Exterior angle of a Regular n-gon**

Polygon Worksheet4. How do you find the measure of one exterior angle of a regular polygon?Measure of One Exterior angle of a Regular n-gon

22 Polygon Worksheet6. Find the measure of an exterior angle of the following regular polygons.DecagonPentagonMeasure of one Exterior AngleMeasure of one Exterior AngleMeasure of one Exterior AngleMeasure of one Exterior AngleMeasure of one Exterior AngleMeasure of one Exterior Angle

23 **Polygon Worksheet The measure of each exterior angle is 30 degrees.**

7. Find the number of sides for a regular polygon using the information given:The measure of each exterior angle is 30 degrees.

24 Polygon Worksheet7. Find the number of sides for a regular polygon using the information given:b) The measure of each interior angle is 135 degrees.*Complete on Board

25 **Polygon Worksheet The measure of each exterior angle is 30 degrees.**

7. Find the number of sides for a regular polygon using the information given:The measure of each exterior angle is 30 degrees.The measure of each interior angle is 135 degrees.

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