Tricky Right Triangle Problem (Picture Included)

**JacobFischer** · October 12th 2012, 01:06 PM

It states that Segments AD, DF, FE, EC, and CB are all equal. Triangle ACB is a Right Triangle. That is all the information given. What is Angle A?
An explanation would be great because I can't seem to understand it!

**MaxJasper** · October 12th 2012, 02:52 PM

Try A=40.6463194 (unchecked

)

**Soroban** · October 12th 2012, 03:49 PM

Hello, JacobFischer!

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Segments AD, DF, FE, EC, and CB are all equal.
Triangle ACB is a Right Triangle.
What is angle A?

Let $\theta$ = angle A.
We will calculate all the angles in the diagram.
Write these figures in your diagram as we go along.

Since $\Delta ADF$ is isosceles, $\angle DFA = \theta.$

Since $\angle EDF$ is an exterior angle, $\angle EDF = 2\theta.$

Since $\Delta EDF$ is isosceles, $\angle DEF = 2\theta.$

Then $\angle DFE \:=\:180^o - 2\theta - 2\theta \:=\:180^o - 4\theta$

Hence, $\angle EFC \:=\:180^o - \theta - (180^o-\theta)\:=\:3\theta$

Since $\Delta FEC$ is isosceles, $\angle ECF = 3\theta.$

Then $\angle FEC \:=\:180^o - 3\theta - 3\theta \:=\:180^o - 6\theta$

Hence, $\angle BEC \:=\:180^o - 3\theta - (180^o - 6\theta) \:=\:4\theta$

Since $\Delta ECB$ is isosceles, $\angle EBC = 4\theta$

Hence, $\angle ECB \:=\:180^o - 4\theta - 4\theta \:=\:180^o - 8\theta$ .[1]

But we know that: $\angle ECB + \angle ECF \:=\:90^o$
. . That is: . $\angle ECB + 3\theta \:=\:90^o$ .[2]

Substitute [1] into [2]: . $(180^o - 8\theta) + 3\theta \:=\:90^o$

. . and we have: . $\text{-}5\theta \:=\:\text{-}90^o \quad\Rightarrow\quad \boxed{\theta \:=\:18^o}$

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