Mathematics Exam > Mathematics Notes > Additional Topics for IIT JAM Mathematics > Quadrilaterals - General Geometry
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Angles
It is common knowledge that the sum of the angles in a triangle is 180° but how about in polygons with a greater numbers of angles? If we are given a convex polygon with n sides and S is the sum of the measures of the interior angles then S = 180(n - 2).
Example
Find the sum of the measures of the interior angles in an octagon.
The octagon has 8 sides and we plug this value into our formula:
S = 180(8 - 2) = 1080°
Hence the sum of the measures of the interior angles in an octagon is 1080°.
Another thing with convex polygons is that the sum of the measures of the exterior angles is always 360°
m∠A + m∠B + m∠C + m∠D + m∠E + m∠F = 360
Properties of parallelograms
One special kind of polygons is called a parallelogram. It is a quadrilateral where both pairs of opposite sides are parallel.
There are six important properties of parallelograms to know:
1. Opposite sides are congruent (AB = DC).
2. Opposite angels are congruent (D = B).
3. Consecutive angles are supplementary (A + D = 180°).
4. If one angle is right, then all angles are right.
5. The diagonals of a parallelogram bisect each other.
6. Each diagonal of a parallelogram separates it into two congruent triangles.
△ACD ≅ △ABC
If we have a parallelogram where all sides are congruent then we have what is called a rhombus. The properties of parallelograms can be applied on rhombi.
If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid. The parallel sides are called bases while the nonparallel sides are called legs. If the legs are congruent we have what is called an isosceles trapezoid.
In an isosceles trapezoid the diagonals are always congruent. The median of a trapezoid is parallel to the bases and is one-half of the sum of measures of the bases.
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FAQs on Quadrilaterals - General Geometry - Additional Topics for IIT JAM Mathematics
| 1. What are the properties of a quadrilateral? |  |
Ans. A quadrilateral is a polygon with four sides and four vertices. The properties of a quadrilateral include:
- The sum of the interior angles is always equal to 360 degrees.
- Opposite angles are equal.
- The opposite sides are parallel in a parallelogram.
- The diagonals bisect each other in a parallelogram.
| 2. What are the types of quadrilaterals? | |
Ans. There are different types of quadrilaterals, including:
- Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
- Rectangle: A parallelogram with all angles equal to 90 degrees.
- Square: A parallelogram with all sides equal in length and all angles equal to 90 degrees.
- Rhombus: A parallelogram with all sides equal in length.
- Trapezoid: A quadrilateral with one pair of opposite sides parallel.
| 3. How do you find the area of a quadrilateral? | |
Ans. The area of a quadrilateral can be found using different formulas depending on the type of quadrilateral:
- Parallelogram: The area is given by base times height.
- Rectangle: The area is given by length times width.
- Square: The area is given by side length squared.
- Rhombus: The area is given by the product of the diagonals divided by 2.
- Trapezoid: The area is given by the average of the lengths of the parallel sides multiplied by the height.
| 4. Can a quadrilateral have all sides equal in length but not all angles equal? | |
Ans. No, a quadrilateral with all sides equal in length is a special type called a square. In a square, all angles are also equal to 90 degrees. If a quadrilateral has all sides equal but not all angles equal, it would be a rhombus.
| 5. What is the sum of the angles in a quadrilateral? | |
Ans. The sum of the angles in a quadrilateral is always equal to 360 degrees. This means that if you add up all the interior angles of a quadrilateral, the total sum will be 360 degrees.
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