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SPECIAL PARALLELOGRAMS (ADDITIONAL EXAMPLES)
TEd Teacher Ed
Overview
This video explains the properties of special parallelograms: rectangles, rhombuses, and squares. It starts by reviewing the general properties of all parallelograms, such as opposite sides and angles being congruent, and diagonals bisecting each other. It then details the unique properties of each special parallelogram: rectangles have four right angles, rhombuses have four congruent sides, and squares possess all properties of both rectangles and rhombuses. The video uses examples to demonstrate how to calculate perimeters, areas, and angle measures within these shapes, reinforcing the application of their specific geometric rules.
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Chapters
- Parallelograms are defined by having two pairs of parallel sides.
- All parallelograms share properties like congruent opposite angles, congruent opposite sides, supplementary consecutive angles, and diagonals that bisect each other.
- Special parallelograms (rectangles, rhombuses, squares) inherit all these general parallelogram properties.
Understanding the foundational properties of parallelograms is crucial because special types of parallelograms are essentially enhanced versions of a basic parallelogram, sharing its core characteristics.
The video lists properties 1-5 as general parallelogram properties applicable to all special parallelograms.
- Rectangles are parallelograms with four right angles (90 degrees).
- In a rectangle, diagonals are congruent (equal in length).
- The perimeter of a rectangle is calculated by summing all its sides, using the property that opposite sides are congruent.
- Angles within a rectangle can be determined using the fact that all angles are 90 degrees and properties of triangles formed by diagonals.
Knowing rectangle-specific properties allows for precise calculations of its dimensions, angles, and perimeter, which are common in real-world applications like construction and design.
Given a rectangle with side FE = 4.5 and side EL = 16.5, the perimeter is calculated as 4.5 + 4.5 + 16.5 + 16.5 = 42.
- Rhombuses are parallelograms with four congruent sides.
- In a rhombus, diagonals are perpendicular bisectors of each other, forming 90-degree angles at their intersection.
- Rhombus diagonals bisect the opposite angles, dividing them into two equal parts.
- The perimeter of a rhombus is simply four times the length of one side, as all sides are equal.
The unique properties of a rhombus, particularly its perpendicular diagonals and angle bisectors, are key to solving problems involving its internal angles and symmetry.
If a rhombus has a side length of 23.5, its perimeter is 4 * 23.5 = 94. If one of the angles formed by the diagonals is 90 degrees, this is a direct application of the perpendicular diagonals property.
- A square is a special parallelogram that combines the properties of both rectangles and rhombuses.
- It has four congruent sides (like a rhombus) and four right angles (like a rectangle).
- The area of a square is calculated by squaring the length of one side (side * side).
- Like rhombuses, the diagonals of a square are perpendicular and bisect the angles they connect, resulting in 45-degree angles within the triangles formed by the diagonals.
Understanding the square as a combination of rectangle and rhombus properties simplifies problem-solving, as you can apply rules from both categories.
For a square with side length 25, the area is 25 * 25 = 625 square units. The diagonals intersect at 90 degrees, and each diagonal divides a vertex angle into two 45-degree angles.
Key takeaways
- All special parallelograms (rectangles, rhombuses, squares) are also parallelograms and share their fundamental properties.
- Rectangles are distinguished by their four right angles and congruent diagonals.
- Rhombuses are defined by their four equal sides and diagonals that bisect angles and are perpendicular.
- Squares possess all the properties of both rectangles and rhombuses, making them highly symmetrical.
- The perimeter of any parallelogram is the sum of its four sides; for special parallelograms, this simplifies due to congruent sides.
- Area calculations differ: for rectangles, it's length times width; for squares, it's side squared.
- Understanding how diagonals behave (bisecting, being congruent, being perpendicular, bisecting angles) is key to solving problems with special parallelograms.
Key terms
ParallelogramRectangleRhombusSquareCongruent sidesCongruent anglesSupplementary anglesDiagonalsBisectPerpendicularPerimeterArea
Test your understanding
- What are the two defining properties of a rectangle that distinguish it from a general parallelogram?
- How do the diagonals of a rhombus differ from the diagonals of a rectangle?
- Why is a square considered a special case of both a rectangle and a rhombus?
- How can you calculate the perimeter of a rhombus if you only know the length of one side?
- What is the relationship between the diagonals of a square, and how does this affect the angles formed at their intersection?