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Math Help - More angles

  1. #1
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    Talking More angles

    In the right-angled triangle ABC opposite, CAB = 90 degrees
    The bisector of ABC meets AC at D.
    Let ABD = z, ACB = y and ADB = x
    Give reasons why 2z = 90 degrees - y and x = z + y
    Hence show that x = 45 degrees + 1/2 y
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  2. #2
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    Looking at angle B, you see that it is equal to 2z. Looking at the triangle ABC, we have the 3 angles of the triangle: y, 2z, and 90 (this is angle CAB). We know that the sum of a triangle's angles add up to 180, so: y + 2z + 90 = 180
    Solve for 2z.

    To prove x = 2z + y, note from the first question:
    2z = 90 - y \quad \Rightarrow \quad 90 = 2z + y.

    Looking at triangle ABD, we see that  x + z = {\color{red}90} . Using the fact that {\color{red}90 = 2z + y}, substitute it in and we get:
    x + z = {\color{red} 2z + y}

    Solve for x.

    For the last one, from the first question we have that 2z = 90 - y \Rightarrow z = 45 - \frac{1}{2}y. Substitute that into the expression for x in the 2nd question and you should be done proving it.
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