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Altitudes AD and BE of acute triangle ABC intersect at point H. If angle ABH = 30 and angle DAC = 45, then what is angle HCA in degrees?

 Jul 8, 2024
 #1
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In order to solve this problem, let's analyze somethings about it. 

First, let's note that right triangles \(ABC, ABE, ABD, BHD, BEC\) are all 30-60-90 triangles. 

Also, note that right triangles \( ABE,ADC \) are 45-45-90 triangles. 

 

In order to solve the problem, let's draw a line to help find HCA for us. 

Let's draw the altitude of ABC by connecting c to line segment AB. 

 

This creates a right triangle!

 

Since we know that

\(\angle BAC =60^\circ\)

 

HCA would have to be 30 degrees because of the right triangle.

So we have

\(\angle HCA = 30^\circ\)

 

So 30 degrees is our answer. 

 

Thanks! :)

 Jul 8, 2024

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