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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?

 May 1, 2024

Best Answer 

 #1
avatar+202 
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Let's start with what we should do for any geometry question, which is to of course, graph it. 

 

Note that we have right triangles ADC, ABE, ABD, BHD, BEC and triangles ABE and ADC are a 30-60-90 and a 45-45-90 respectively. 

 

Now, it's pretty simple from here. 

 

We first draw the altitude for triangle ABC from point C to line segment AB, it intersects point H along the way.

 

Also, since we know BAC is 60 degrees, HCA would be 30 degrees!

 

30 degrees!

 

Thanks!

 May 1, 2024
 #1
avatar+202 
+1
Best Answer

Let's start with what we should do for any geometry question, which is to of course, graph it. 

 

Note that we have right triangles ADC, ABE, ABD, BHD, BEC and triangles ABE and ADC are a 30-60-90 and a 45-45-90 respectively. 

 

Now, it's pretty simple from here. 

 

We first draw the altitude for triangle ABC from point C to line segment AB, it intersects point H along the way.

 

Also, since we know BAC is 60 degrees, HCA would be 30 degrees!

 

30 degrees!

 

Thanks!

NotThatSmart May 1, 2024

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