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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 in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Proof: In quadrilateral ABCD
∠A = ∠D
`=>` ∠A +∠B +∠C + ∠D
Now, ∠A +∠B +∠C + ∠D = 360° (angle sum property of quadrilateral)
`=>` 2(∠A +∠B ) = 360°
`=> ` ∠A +∠B = 180°
`therefore` ∠A +∠B = ∠C + ∠D = 180°
Line AB intersects AD and BC at A and B respectively.
Such that ∠A +∠B = 180°
`therefore` AD || BC (Sum of consecutive interior angle is 180° ) ...(1)
∠A +∠B = 180°
∠A +∠D = 180° (∠B= ∠D)
`therefore` AB || CD ..(2)
From (1) and (2), we get
AB || CD and AD || BC
`therefore` ABCD is a parallelogram.
Shaalaa.com | Theorem : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
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