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- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
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- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Another Condition for a Quadrilateral to Be a Parallelogram
- The Mid-point Theorem
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- 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
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notes
The opposite angles of a parallelogram are of equal measure.
Given: ABCD is a parallelogram.
To Prove: m∠ A = m∠C
Construction: Draw diagonal AC and diagonal BD.
To Prove:
AC and BD are the diagonals of the parallelogram,
∠ BAC = ∠ ACD and ∠ DAC = ∠ ACB......(pair of alternate angles) and `bar(AC)` is common side,
In ΔABC and ∆ADC,
∠ BAC = ∠ ACD ......(pair of alternate interior angles)
∠ DAC = ∠ ACB ......(pair of alternate interior angles)
Side AC = Side AC ......(common side)
∆ABC ≅ ∆CDA ......(ASA congruency criterion)
This shows that ∠B and ∠D have same measure. In the same way you can get, m∠A = m ∠C.
Alternatively, ∠ BAC = ∠ ACD and ∠ DAC = ∠ ACB
We have, m∠ A = ∠ BAC + ∠ DAC and ∠ ACB + ∠ ACD = m∠C.
m∠A = m ∠C.
Hence Proved.
Example
In the given Fig, BEST is a parallelogram. Find the values x, y, and z.
S is opposite to B.
So,
x = 100° ........(opposite angles property)
x = 100° ........(opposite angles property)
y = 100° ..........(a measure of angle corresponding to ∠ x)
z = 80° ..........(since ∠y, ∠z is a linear pair)
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