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# Geometry

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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?

Jun 2, 2024

#1
+806
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First, let's note somethings.

Notice that right triangles ABC, ABE, ABD, BHD, BEC are 30-60-90 triangles.

Right triangles ABe and ADC are 45-45-90 triangles.

Now, let's draw the altitude for triangle ABC by connecting point C to line segment AB.

This line intersects point H on the way.

We know that BAC is is 60 degrees, so that leaves HCA would be 30 degrees.

So 30 degrees is our answer!

Thanks! :)

Jun 2, 2024

#1
+806
+1

First, let's note somethings.

Notice that right triangles ABC, ABE, ABD, BHD, BEC are 30-60-90 triangles.

Right triangles ABe and ADC are 45-45-90 triangles.

Now, let's draw the altitude for triangle ABC by connecting point C to line segment AB.

This line intersects point H on the way.

We know that BAC is is 60 degrees, so that leaves HCA would be 30 degrees.

So 30 degrees is our answer!

Thanks! :)

NotThatSmart Jun 2, 2024