In ∆ABC acute angles are in the ratio 5:1, i. e.

∠BAC : ∠ABC = 5:1. If CH is an altitude and

CL is an angle bisector, find m∠HCL.

Let x be the measure of the angle A and y be the measure of the angle B.

Since ABC is a right angled triangle, then:

x+y = 90 degrees.

Also, x/y = 5 (Given)

therefore x=5y. Using the above equation again like this:

5y+y=90 then 6y=90 and then y=15.

So x=5*15=75.

The measure of the angle A is 75 degrees.

Since the triangle AHC is right angled, then ACH = 90-75=15 degrees.

Also, since CL is a bisector, then ACL=90/2=45.

Now, HCL=45-15=30.