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Question
The sides BA and DC of a quadrilateral are produced as shown in the given figure. Prove that x+y=a+b.
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Solution
We know that the internal angle at A and ∠b form a linear pair of angles.
Therefore, we can say that ∠A + ∠b = 180°.
The measure of internal angle at A would be “∠A = 180° - b” at ∠A.
Similarly, the internal angle at C and ∠a form a linear pair of angles. So, we can say the measure of the internal angle at C would be “∠C = 180° - a” at ∠C.
Now that we have all 4 internal angles, we can now make use of angle sum property of a quadrilateral.
Since we know that they will add up to 360°, we can write:
∠A + ∠B + ∠C + ∠D = 360°
180° - b + x + 180° - a + y = 360°
360° + x + y - (a + b) = 360°
By subtracting 360° from both sides, we get
x + y - (a + b) = 0
x + y = a + b
Hence proved.
Therefore, we can say that ∠A + ∠b = 180°.
The measure of internal angle at A would be “∠A = 180° - b” at ∠A.
Similarly, the internal angle at C and ∠a form a linear pair of angles. So, we can say the measure of the internal angle at C would be “∠C = 180° - a” at ∠C.
Now that we have all 4 internal angles, we can now make use of angle sum property of a quadrilateral.
Since we know that they will add up to 360°, we can write:
∠A + ∠B + ∠C + ∠D = 360°
180° - b + x + 180° - a + y = 360°
360° + x + y - (a + b) = 360°
By subtracting 360° from both sides, we get
x + y - (a + b) = 0
x + y = a + b
Hence proved.
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