Altitudes AD and BE of triangle ABC intersect at H . If angle AHB = 120 degrees and angle BAC = 52 degrees, then what is angle AHC?

siIviajendeukie Jun 25, 2024

#1**+1 **

I'm not sure if my logic is correct, but I'll give it a try.

The point where the 3 altitudes of a triangle meet is the orthocenter. H is the orthocenter in this problem.

Now, let's extend segment CH until it meets AB. Name this point F.

CF is the altitude now.

Let's note that \(\angle CFA = 90^\circ\) since CF is now the altitude.

We can write the equation \(\angle ACF = 180 - 90 - 52\) through triangle CAF, so we have \(\angle ACF = 38\)

Through triangle EHC, we have \(\angle CHE = 180 - 38 - 90 = 52\). Since angle CHE is equal to angle FHB, angle FHB is also 52 degrees.

Now, we can write the equation \(\angle BAC + \angle CFA + \angle BEA + \angle EHF = 360\) since it forms a quadrilateral.

Plugging in what we have, we get that

\(52+90+90+\angle EHF = 360 \\ \angle EHF = 128\)

Now, angle EHF and angle AHB share angle AHF. Since FHB is 52, we have that angle AHF is 120 - 52 = 68 degrees.

EHA is now 128- 68 = 20 degrees.

Thus, angle AHC is equal to 20 + 52 = 72 degrees.

So our answer is 72.

I'm not sure if this is right....

Thanks! :)

NotThatSmart Jun 25, 2024

#2**+1 **

Sorry, there was a slight error in my math.

We have angle EHA as \(128-68=60\) degrees.

Therefore, our answer iss

\(60+52=112\) degrees.

Still not sure if this is right...

Thanks! :)

NotThatSmart
Jun 25, 2024