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Introduction to Euclid’S Geometry
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- Introduction to Lines and Angles
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- 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
Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
proof : 
In ∆ AOD and ∆ COB (Given)
OD = OB (Given)
∠AOB = ∠COD (Vertically opposite angles are equal)
Therefore , ∆AOD ≅ ∆COB (By SAS criterion of congruence)
So, ∠OAD = ∠OCB ...(1) (C.P.C.T)
Now , lines AC intersects AD and BC at A and C respectively,
such that ∠OAD = ∠OCB ...[from(1)]
`therefore` ∠OAD and ∠ OCB form a pair of alternate interior angles are equal.
Thus, AD || BC
Similarly , we can prove that AB || DC
Hence , quadrilatera; ABCD is a parallelogram.
Shaalaa.com | Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
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