angle abc is an equilateral triangle with squares constructed on sides ab and ac is shown in the given image find the value of angle x
Answers
Given that ABC is an equilateral triangle.
Therefore, ∠ABC=∠CBA=∠BAC=60
o
GIven that ABXW and AYZC are two squares.
Therefore, AX is a diagonal of square ABXW - as shown in the figure.
All interior angles of a square are equal to 90
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Therefore, ∠BAW=90
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The diagonal of a square bisects the angle at the vertex.
Therefore, ∠BAX=∠AXW=45
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⇒∠OAX=45
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-------(1)
Join vertices X and Z - as shown in the figure. Join ZX, a straight line.
Line ZX cuts the sides AB and AC, of the Equilateral triangle ABC, at O and N respectively.
The smaller triangle AON is similar to triangle ABC are similar, thus triangle AON is also an equilateral triangle.
Therefore, ∠NAO=∠AON=∠ANO=60
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Line A0 is cutting the straight line XZ, hence
∠ZOA+∠XOA=∠ZOX=180
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60
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+∠XOA=180
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∠XOA=180
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−60
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∠XOA=120
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-------(2)
Consider triangle AOX - in figure:
Here, ∠XOA+∠OAX+∠OXA=180
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120
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+45
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+∠OXA=180
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, from equations (1) and (2)
∠OXA=180
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−120
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−45
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∠OXA=15
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∠OXA=∠ZXA=15
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Therefore,
10
1
∠ZXA=
10
1
×15
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10
1
∠ZXA=1.5
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