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# Triangle Question

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Altitudes AX and BY of acute triangle ABC intersect at H. If angle BAC = 43 and angle ABC = 67, then what is the angle HCA? Thank You :P

Nov 4, 2018

#1
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Angle BAC = angle ZAC  =  43°

Angle CZA = angle CZB = 90°

So triangle ZAC  is a right triangle and angle  ZCA  is complementary to angle ZAC  = 47°

And angle ZCA   =   angle HCA  =   47°

EDIT TO CORRECT AN ERROR>>>>!!!!   Nov 4, 2018
edited by CPhill  Nov 4, 2018
#2
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CPhill I think you have a mistake,all it's right but in the end it's

angle ZCA   =   angle HCA  =   47° NOT   43°

because :

angle CZA + angle ZAC + angle ZCA = 180° in triangle ZAC

so 43° + 90° + angle ZCA =  180°

so angle ZCA = 180°-133° = 47°

So angle ZCA   =   angle HCA  =   47°

Dimitristhym  Nov 4, 2018
edited by Dimitristhym  Nov 4, 2018
edited by Dimitristhym  Nov 4, 2018
#3
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Oops!!!....just a typo.....you are correct!!!

I have fixed it....   CPhill  Nov 4, 2018
edited by CPhill  Nov 4, 2018
#4
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Thanks!

CoolStuffYT  Nov 6, 2018