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Question
In the given figure, if BP || CQ and AC = BC, then the measure of x is
Options
20°
25°
30°
35°
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Solution
In the given figure, BP || CQ and AC || BC
We need to find the measure of x
Here, we draw a line RS parallel to BP, i.e BP || RS
Also, using the property, “two lines parallel to the same line are parallel to each other”
As,
BP || RS
BP || CQ
Thus, RS || CQ
Now, BP || RS and BA is the transversal, so using the property, “alternate interior angles are equal”
∠PBA = ∠BAS
∠BAS = 20° .......... (1)
Similarly, CQ|| RS and AC is the transversal
∠QCA = ∠SAC
∠SAC = x ........(2)
Adding (1) and (2), we get
∠SAC + ∠BAS = 20° + x
∠A = 20° + x
Also, as AC = BC
Using the property,”angles opposite to equal sides are equal”, we get
∠ABC = ∠CAB
∠ABC = 20° + x
Further, using the property, “an exterior angle is equal to the sum of the two opposite interior angles”
In ΔABC
ext. ∠C = ∠CAB + ∠ABC
70° + x = 20° + x + 20° + x
70° + x = 40° + 2x
70° - 40° = 2x - x
x = 30°
Thus, x = 30°
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