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Properties of a Parallelogram - Property: The Opposite Angles of a Parallelogram Are of Equal Measure.

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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)
y = 100° ..........(a measure of angle corresponding to ∠ x)
z = 80°   ..........(since ∠y, ∠z is a linear pair)
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