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Grade 9 Math | Properties of Parallelogram, Find Angles, Sides, Perpendicular Height & Diagonals
Native man Math tutorial
Overview
This video lesson explains the properties of parallelograms and how to apply them to find unknown angles, sides, perpendicular heights, and diagonals. It begins by defining a parallelogram as a quadrilateral with two pairs of parallel sides and briefly mentions rectangles, rhombuses, and squares as types of parallelograms. The lesson then details how to name the parts of a parallelogram, including angles and sides, using single letters, three letters, or numbers. It systematically presents five key properties: opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other, and a diagonal divides the parallelogram into two congruent triangles. Finally, the video demonstrates how to use these properties to solve various problems involving finding missing measures in given parallelograms, including calculations for area and perpendicular height.
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Chapters
- A parallelogram is a quadrilateral with two pairs of parallel sides.
- Rectangles, rhombuses, and squares are specific types of parallelograms.
- Parts of a parallelogram (angles and sides) can be named using single letters, three letters (with the vertex in the middle), or numbers.
- Angles can also be named using lowercase letters, and sides using lowercase letters or line segment notation.
Understanding how to identify and name the components of a parallelogram is fundamental before applying its properties to solve problems.
Naming angle M in parallelogram MATH as angle AMH, ensuring M is the middle letter representing the vertex.
- Property 1: Opposite sides are parallel and congruent (equal in length).
- Property 2: Opposite angles are congruent (equal in measure).
- Property 3: Consecutive angles are supplementary (their sum is 180 degrees).
- Property 4: Diagonals bisect each other (they cut each other into two equal parts).
- Property 5: A diagonal divides the parallelogram into two congruent triangles.
These properties provide the rules and relationships that govern all parallelograms, enabling us to deduce unknown measures from known ones.
In parallelogram MAKE, if angle E is 70°, then its opposite angle A must also be 70°.
- Use the property that opposite sides are congruent to find unknown side lengths.
- Use the property that opposite angles are congruent to find unknown angle measures.
- Use the property that consecutive angles are supplementary to find unknown angle measures.
- When diagonals intersect, the segments formed are equal due to the bisection property.
This section shows how to practically use the established properties to calculate specific missing values within a parallelogram.
In parallelogram MATH, if angle M is 60°, then angle T (opposite) is also 60°, and angles A and H (consecutive to M) are 180° - 60° = 120°.
- The perpendicular height (h) is the shortest distance between a base and its opposite side, forming a 90° angle.
- The area of a parallelogram is calculated by multiplying the base (b) by its perpendicular height (Area = b * h).
- The formula can be rearranged to find the height if the area and base are known (h = Area / b).
- The formula can also be rearranged to find the base if the area and height are known (b = Area / h).
This knowledge is crucial for solving problems involving the space enclosed by a parallelogram, a common application in geometry and real-world scenarios.
If a parallelogram has an area of 144 sq meters and a base of 16 meters, its perpendicular height is 144 / 16 = 9 meters.
- Set up equations by equating expressions for congruent sides or angles.
- Use the property of opposite sides being congruent to solve for unknown variables in side length expressions.
- Use the property of opposite angles being congruent to solve for unknown variables in angle measure expressions.
- Substitute the solved variable back into the original expressions to find the actual measures of sides or angles.
This integrates algebraic skills with geometric properties, allowing for the solution of more complex problems where measures are represented by variables.
For parallelogram MAHT where MA = 2x - 30 and HT = x + 15, set 2x - 30 = x + 15, solve for x = 45, and then find MA = 2(45) - 30 = 60.
Key takeaways
- A parallelogram is defined by its two pairs of parallel sides, which also implies congruent opposite sides and angles.
- Consecutive angles in a parallelogram always add up to 180 degrees.
- The diagonals of a parallelogram cut each other exactly in half.
- The area of a parallelogram is found by multiplying its base by its perpendicular height.
- Geometric properties of parallelograms can be used to set up and solve algebraic equations for unknown measures.
- Understanding these properties allows for the calculation of any missing side, angle, or diagonal segment if enough information is provided.
Key terms
ParallelogramQuadrilateralParallel sidesCongruent sidesCongruent anglesSupplementary anglesConsecutive anglesDiagonalsBisectPerpendicular heightBaseArea
Test your understanding
- What are the two defining conditions for a quadrilateral to be classified as a parallelogram based on its sides?
- How do the measures of opposite angles and consecutive angles in a parallelogram relate to each other?
- Explain what it means for the diagonals of a parallelogram to 'bisect each other'.
- How can you calculate the perpendicular height of a parallelogram if you know its area and base length?
- If two opposite sides of a parallelogram are given as algebraic expressions (e.g., 2x - 5 and x + 10), how would you find the value of x?