Angle 3 times bigger

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Kris76

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Jan 27, 2020
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ABC is isosceles triangle. AC= BC; DB=BH
Proof angle GFD is 3 times angle DHB
triangle.jpg
 
You might start by marking pairs of equal angles, namely the base angles of each isosceles triangle, and the vertical angles. Maybe call those x and y. Then call angle GFD z, and write two equations relating x, y, and z, using the fact that the sum of angles in a triangle is 180 degrees. Then solve for z by eliminating one of the other angles.

There are other ways. Please show us some attempt, so we can see where you are and guide you to the next step.

Keep in mind that we are here to help you, not to replace you, so you need to be doing the work. See our guidelines.
 
In your title says "Three times bigger" but in the body you say "three times angle DHB". Those are NOT the same thing!
 
HallsofIvy said:
In your title says "Three times bigger" but in the body you say "three times angle DHB". Those are NOT the same thing!
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Sorry if I have been misunderstood .It's my fault. Problem/task is to proof that angle GFD is 3 times bigger than angle DHB
Once again my appologies
 
HallsofIvy said:
So angle GFD is 4 times angle DHB? (And the verb is "prove" not "proof".)
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Sorry for my error.

This my way of solving though I thing there is another way. So, this was the reason. to ask to prove this problem hopefully in other way.
So:
AC=BC; DB=BH given
This means that angle BDH=angle BHD
Let angle BDH be β; angle BHD will be also β
Let angle DBH be α
Angle ADF=angle HDB= β
In triangle DBH 180o = α + 2 β; α= 180-2β
Angle CAB= α -> 180 - 2β
Angle FAB= 180o – α = 180o – (180o - 2 βangle) = 2 β
Angle AFD – 180o – (angle FAB + angle ADF)
Angle AFD = 180o – (2 β + β) = 180o - 3 β
Angle GFD= 180o – angle AFD = 180o – (180o - 3 β)

So Angle GFD = 3 β

I am looking for another way (if any). If knows please show it.
 
I presume that the wording of the title is your own, and you more or less quoted the problem itself, so that the latter is what you were actually told to do.

If so, then the problem (about multiplication by 3) is correct, and your wording in the title is technically wrong. (I find that "three times more" is so commonly used to mean "3 times as large", that I consider it an idiom -- something we say that is not literally correct, but is understood that way in common use.) The fact is, angle GFD is 3 times as large as angle DHB, as you'll find when you carry out the proof!

The important thing is to solve the problem! Please show some work, so we can help you. "Three times bigger" is a distraction.
 
I've added some labels to the angles to help clarify their relationships:
triangle.png​
We want to prove that [MATH]v = 3y[/MATH]. Observe that [MATH]x = 180^{\circ} - z = 2y[/MATH]. Where can we go from here?
 
Mr. Bland said:
I've added some labels to the angles to help clarify their relationships:
We want to prove that [MATH]v = 3y[/MATH]. Observe that [MATH]x = 180^{\circ} - z = 2y[/MATH]. Where can we go from here?
Click to expand...
Well, most probably there is something unclear for you.

Proof is clear and clean
x = 2y; angle ADF = y
w =180 – (x +y) or w = 180 – (2y +y) = 180 – 3y
v = 180 – w or v = 180 – (180 - 3y) = 180 – 180 – (-3y) = 3y

Finally v (or angle GFD) = 3y (or angle BDH)
 
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