Quadrilaterals
A quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.
Summary
A quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.
Things to Remember
- Quadrilateral just means 'four sides' (quad means four, lateral means side).
- Quadrilateral are simple (not self-intersecting) or complex (self- intersecting) and also called crossed.
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Quadrilaterals
A closed plane figure formed by four line segments is a quadrilateral. A quadrilateral is also known as a polygon with four sides and four vertices or corners.
A quadrilateral ABCD has
- four sides: AB, BC, CD and DA
- four angles:∠A, ∠B,∠C and ∠D
There are many kinds of quadrilaterals. Such as:
1. Parallelogram
Quadrilaterals having opposite sides parallel is known as a parallelogram.
In the figure AB ⁄⁄ CD and AD ⁄⁄ BC. So, ABCD is a parallelogram.
Theorems with parallelogram:
Theorem 1
The opposite sides of a parallelogram are congruent.
Verification:
Draw three parallelograms of different sizes as shown below:
Measure the sides and complete the table below:
| Figure | WZ | XY | Result | WX | ZY | Result |
| (i) | WZ=XY | WX = ZY | ||||
| (ii) | ||||||
| (iii) |
Conclusion: Opposite sides of a parallelogram are equal.
Theorem 2
The opposite angles of a parallelogram are congruent.
Verification:
Draw three parallelograms of different sizes.
Measure the opposite angles and complete the table below:
| Figure | ∠W | ∠Y | Result | ∠X | ∠Z | Result |
| (i) | ∠W =∠Y | ∠X =∠Z | ||||
| (ii) | ||||||
| (iii) |
Conclusion: The opposite angles of a parallelogram are congruent.
Theorem 3
The diagonals of a parallelogram bisect each other.
Verification:
Draw three parallelograms of different sizes. Join the diagonals WY and XZ.
Measure the segments of the diagonals and complete the table below:
| Figure | WO | YO | Result | XO | ZO | Result |
| (i) | WO = YO | XO =ZO | ||||
| (ii) | ||||||
| (iii) |
Conclusion: Diagonals of the parallelogram bisect each other.
2. Rectangle
The rectangle is a parallelogram with all angles 90o. Opposite sides are parallel and of equal length. It is also known as an equiangular parallelogram.
Diagonal created in a rectangle are also congruent.
Theorem
The diagonals of a rectangle are congruent.
Verification:
Draw three rectangles of different sizes. Join the diagonals WY and XZ.
Measure the diagonals WY and XZ with the ruler and complete the following table.
| Figure | WX | XZ | Result |
| (i) | WX = XZ | ||
| (ii) | |||
| (iii) |
Conclusion: The diagonals of the rectangle are congruent.
3. Square
Square is also a parallelogram with all sides and angles equal. It is also known as an equilateral and equiangular parallelogram. In another word, a square is a rectangle having adjacent sides equal. The diagonal of square bisects each other at right angles.
Theorem
The diagonals of a square bisect each other at right angles.
Verification:
Draw three squares of different sizes. Join the diagonals WY and XZ which intersect at O. Since a square is a parallelogram, the diagonals bisect each other i.e WO =YO and XO = ZO.
Measure the angles between the diagonals and complete the following table.
| Figure | ∠WOX | ∠YOZ | ∠WOZ | ∠XOY | Result |
| (i) | ∠WOX =∠YOZ =∠WOZ =∠XOY = 90° | ||||
| (ii) | |||||
| (iii) |
Conclusion: The diagonals of a square bisect each other at right angles.
Lesson
Geometry
Subject
Compulsory Maths
Grade
Grade 8
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