Topics
Number Systems
Number Systems
Polynomials
Algebra
Algebraic Expressions
Algebraic Identities
Coordinate Geometry
Linear Equations in Two Variables
Coordinate Geometry
Geometry
Area
Constructions
- Introduction of Constructions
- Geometric Constructions
- Some Constructions of Triangles
Introduction to Euclid’S Geometry
Mensuration
Statistics and Probability
Lines and Angles
- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Concept of Pairs of Angles
- Concept of Transversal Lines
- Basic Properties of a Triangle
Probability
Triangles
Quadrilaterals
- Properties of Quadrilateral
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
Circles
Areas - Heron’S Formula
- Area of a Triangle by Heron's Formula
- Application of Heron’s Formula in Finding Areas of Quadrilaterals
- Geometric Interpretation of the Area of a Triangle
Surface Areas and Volumes
Statistics
Theorem
The opposite sides of a parallelogram are of equal length.
Given: ABCD is a parallelogram.
To Prove: AB = DC and BC = AD.
Construction: Draw any one diagonal, say `bar(AC)`.
Proof:
Consider a parallelogram ABCD,
In triangles ΔABC and ΔADC,
∠ 1 = ∠2, ∠ 3 = ∠ 4 .....(Pair of alternate angle)
and `bar(AC)` is common side.
Side AC = Side AC .....(common side)
∠ 1 ≅ ∠2 .....(Pair of alternate angle)
∠ 3 ≅ ∠ 4 .....(Pair of alternate angle)
by ASA congruency condition,
∆ ABC ≅ ∆ CDA
This gives AB = DC and BC = AD.
Hence Proved.
Example
Find the perimeter of the parallelogram PQRS.
In a parallelogram, the opposite sides have the same length.
Therefore, PQ = SR = 12 cm and QR = PS = 7 cm.
So,
Perimeter = PQ + QR + RS + SP
= 12 cm + 7 cm + 12 cm + 7 cm
= 38 cm.
Shaalaa.com | To Prove that the Opposite Sides of a Parallelogram are of Equal length
to track your progress
Series: Property: the Opposite Sides of a Parallelogram Are of Equal Length.
0%
