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SOLVING PROBLEMS INVOLVING SIDES AND ANGLES OF A POLYGON || GRADE 7 MATHEMATICS Q3
WOW MATH
Overview
This video explains how to solve problems involving the sides and angles of polygons, focusing on Grade 7 Mathematics. It begins with a review of interior and exterior angles, especially for regular polygons, and introduces formulas for calculating the sum and measure of individual interior and exterior angles based on the number of sides. The video then walks through various examples, including finding unknown angles using properties like supplementary angles, vertical angles, and the exterior angle theorem. It also covers problems related to quadrilaterals, pentagons, hexagons, heptagons, isosceles trapezoids, and parallelograms, demonstrating how to set up and solve equations to find unknown angle measures and side lengths.
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Chapters
- In regular polygons, all interior angles are congruent, and all exterior angles are congruent.
- The sum of interior angles of an n-sided polygon is (n-2) * 180 degrees.
- The measure of each interior angle of a regular n-sided polygon is ((n-2) * 180) / n degrees.
- An interior angle and its adjacent exterior angle at the same vertex are supplementary (sum to 180 degrees).
Understanding these fundamental properties is crucial for calculating unknown angles and sides in any polygon, forming the basis for solving more complex problems.
A table showing the number of sides, number of triangles formed, sum of interior angles, and measure of each interior angle for polygons from triangles to decagons.
- Exterior angles can be identified by their position outside the polygon.
- Remote interior angles are the two interior angles not adjacent to a given exterior angle.
- The exterior angle theorem states that an exterior angle is equal to the sum of its two remote interior angles.
- Vertical angles are opposite angles formed by intersecting lines and are always congruent.
- Angles forming a linear pair (adjacent angles on a straight line) are supplementary.
These relationships allow us to find the measure of unknown angles by using the known angles and the geometric properties of polygons and intersecting lines.
Given an exterior angle of 125 degrees and one remote interior angle of 68 degrees, find the other remote interior angle by subtracting: 125 - 68 = 57 degrees.
- For quadrilaterals (4 sides), the sum of interior angles is 360 degrees.
- For pentagons (5 sides), the sum of interior angles is 540 degrees.
- For hexagons (6 sides), the sum of interior angles is 720 degrees.
- For heptagons (7 sides), the sum of interior angles is 900 degrees.
- Unknown angles can be found by setting up an equation using the sum of interior angles for the specific polygon.
This demonstrates how to apply the general formulas to polygons with different numbers of sides, solving for missing angle measures by creating algebraic equations.
In a pentagon where angles are represented as expressions involving 'x', sum them up, set the sum equal to 540, solve for 'x', and then substitute 'x' back into each angle's expression.
- In an isosceles trapezoid, base angles are congruent.
- Properties of parallel lines cut by a transversal (alternate interior angles are congruent) can be used.
- The perimeter of a polygon is the sum of all its side lengths.
- If some sides are congruent, this information can be used in perimeter calculations.
This section shows how to integrate angle properties with side length calculations and specific quadrilateral properties like those of isosceles trapezoids.
For an isosceles trapezoid with angles represented as 3x, 3x, 6x, and 6x, set their sum to 360 degrees and solve for x to find the angle measures.
- Problems can involve finding unknown angles in triangles by using the 180-degree sum rule.
- Missing angles in polygons can be expressed in terms of ratios, requiring algebraic solutions.
- The relationship between an interior angle and an exterior angle can be used to find the number of sides of a regular polygon.
- Consecutive angles in a parallelogram are supplementary.
These examples illustrate more complex scenarios, including ratio problems and word problems that require careful translation into geometric equations.
If one interior angle of a regular polygon is 60 degrees more than twice the exterior angle, set up the equation (2x + 60) + x = 180, solve for the exterior angle x, and use the formula n = 360/x to find the number of sides.
Key takeaways
- Regular polygons have congruent interior and exterior angles, simplifying calculations.
- The sum of interior angles formula ((n-2) * 180) is fundamental for solving angle problems in any polygon.
- Understanding relationships like supplementary, vertical, and remote interior angles is key to finding unknown angle measures.
- The exterior angle theorem provides a direct link between an exterior angle and two interior angles.
- Algebraic skills are essential for solving polygon problems, as unknown angles are often represented by variables.
- Special quadrilaterals like isosceles trapezoids and parallelograms have unique properties that can be applied to problems.
- The perimeter of a polygon is simply the sum of its side lengths, useful when some sides are unknown or congruent.
Key terms
PolygonRegular PolygonInterior AngleExterior AngleSum of Interior AnglesCongruentSupplementary AnglesVertical AnglesRemote Interior AnglesExterior Angle TheoremQuadrilateralPentagonIsosceles TrapezoidParallelogramPerimeter
Test your understanding
- How does the sum of interior angles change as the number of sides of a polygon increases?
- What is the relationship between an interior angle and its adjacent exterior angle in a regular polygon, and why is this important?
- How can you use the exterior angle theorem to find an unknown interior angle if you know the exterior angle and one remote interior angle?
- Explain the steps to find the measure of each angle in a quadrilateral if the angles are given as expressions involving a variable 'x' and the sum of interior angles is 360 degrees.
- What properties of an isosceles trapezoid can be used to solve for its unknown angles?