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Problem-Solving with Angles in Polygons

June 22, 2024.

Problem-solving with angles in polygons is the focus of this mathematics lesson, where students begin by learning to calculate the interior and exterior angles, starting with those of a pentagon. As the lesson progresses, they advance to tackling the angles of composite regular polygons.

Problem-Solving with Angles in Polygons Differentiated Learning Objectives

• All students should be able to calculate an exterior angle for a regular polygon.
• Most students should be able to calculate an interior and exterior angle for a regular polygon.
• Some students should be able to calculate the number of sides for a regular polygon when given its interior or exterior angle.

Questions to check understanding

Basic Understanding: “What are the properties of an isosceles triangle that can help us calculate angle w?”

Application: “Given that the squares are congruent, what can we say about the angles within the squares, and how does that help us determine the value of w?”

Analytical Thinking: “Can anyone explain why the angle sum of a triangle is 180∘ and how this applies to the isosceles triangle shown in the diagram?”

Problem-Solving: “If one angle in the isosceles triangle is 65∘, what are the measures of the other two angles, and what does this tell us about angle w?”

Reasoning Extension: “How would the value of w change if this were not an isosceles triangle but a different type of triangle, and why?”

Problem-Solving with Angles in Polygons Teaching and Development Phase

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Common Misconceptions

Confusing Interior and Exterior Angles: Students may confuse interior angles with exterior angles.

Misunderstanding Angle Sums: For exterior angles, students might not realize that they always sum up to 360 degrees for any polygon and the interior and exterior angles of a polygon lie along a straight line.

Overlooking Supplementary Angles: When polygons are within other shapes, like the examples in the provided problems, the students might miss that some angles are supplementary to each other (they add up to 180 degrees), especially when the polygons share sides or vertices.

Ignoring Given Conditions: Problems often provide specific conditions, such as regularity (all angles and sides are equal) or congruency (polygons are identical in shape and size). Not applying these conditions when calculating angles can lead to incorrect answers.

Problem-Solving with Angles in Polygons Plenary

To support students with the plenary the teacher could:

• Introduce and Recap Key Concepts : Prompt students with questions like “What do you remember about the angles in a regular polygon?” and “How can we calculate the sum of interior angles?”
• Problem Analysis : Encourage students to describe what they see in the diagram. Ask “What shapes are involved?” and “What does it mean for two polygons to be congruent?” This helps them to engage with the problem context.
• Angle Calculation : Work together to find the sum of interior angles of a heptagon, and then calculate the measure of one interior angle. Ask “If we know one interior angle, how can we find its corresponding exterior angle?”
• Identify Supplementary Angles : Guide students to identify where supplementary angles are formed in the diagram and how they relate to the angles of the heptagons and the rectangle.
• Apply and Solve : Prompt students with “How do the angles we’ve calculated help us to find the shaded angles?” and “Can you explain the steps you took to solve for the shaded angles?”

Challenge Questions for Most Able Students

This pattern is made from a regular nine-sided polygon surrounded by squares.

Work out the value of angle a shown in the red region.

Worked Solution

The diagram shows three congruent pentagons and a triangle.

Work out the simplified ratio of angle a to angle b.

Key Points to Remember

• Exterior Angles Sum to 360 Degrees : For any polygon, the sum of the exterior angles, one at each vertex, is always 360 degrees. This is true regardless of the number of sides. When a problem involves exterior angles, remember that dividing 360 degrees by the number of sides will give you the measure of each exterior angle in a regular polygon.
• Interior and Exterior Angles are Supplementary : In a polygon, each pair of interior and exterior angles (at the same vertex) are supplementary if the exterior angle is extended from the interior angle. This means they add up to 180 degrees. This relationship can often be used to find unknown angle measures, especially when dealing with problems that present only a portion of the polygon or when polygons are nested within other shapes.
• Regular vs. Irregular Polygons : When dealing with more complex polygon problems, it’s crucial to distinguish between regular and irregular polygons. A regular polygon has all sides and all angles equal, which allows for the use of symmetry in problem-solving. However, with irregular polygons, you cannot make this assumption, and more information is typically needed to solve for specific angles.

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