Angle between diagonals of an irregular pentagon

Question

In pentagon $ABCDE$, $DC$ is parallel to $BE$ and $BC$ is parallel to $AE$. $△DCE$ and $△BCE$ are both isoscles with $$∠CBE= ∠DEC = α,\ ∠EAB= α+\frac{π}{2}.$$ Suppose $AD$ meets $BE$ at $G$. Find $∠BGA$.

I have tried producing $AD$ to meet $BC$ at $F$ and showing that $∠DFC$ must be $α$, but I haven't been able to show definitively a value for $∠BGA$, which I think is $2α$. I have also tried producing lines from $D$ to $BC$ and from $A$ to $BC$ both inclined at an angle $α$ to $BC$, but I haven't been able to show that they are the same line.

Any hints about how to proceed would be much appreciated!

Answer 1

Let $DC=ED=a$.

Hence, $$EC=CB=2a\cos\alpha$$ and $$EB=2EC\cos\alpha=4a\cos^2\alpha.$$ Thus, by law of sines for $\Delta EAB$ we obtain: $$\frac{EA}{\sin\left(90^{\circ}-2\alpha\right)}=\frac{4a\cos^2\alpha}{\sin\left(90^{\circ}+\alpha\right)},$$ which gives $$EA=4a\cos\alpha\cos2\alpha.$$ Now, since $\alpha<60^{\circ}$, by law of cosines for $\Delta DEA$ we obtain: $$DA=\sqrt{a^2+16a^2\cos^2\alpha\cos^22\alpha-8a^2\cos\alpha\cos2\alpha\cos3\alpha}=$$ $$=a\sqrt{1+8\cos\alpha\cos2\alpha(2\cos\alpha\cos2\alpha-\cos3\alpha)}=a\sqrt{1+8\cos^2\alpha\cos2\alpha}=$$ $$=a\sqrt{1+4(1+\cos2\alpha)\cos2\alpha}=a\sqrt{(1+2\cos2\alpha)^2}=a(1+2\cos2\alpha).$$ Thus, by law of cosines again we obtain: $$\cos\measuredangle DAE=\frac{AE^2+AD^2-DE^2}{2AE\cdot AD}=\frac{16a^2\cos^2\alpha\cos^22\alpha+a^2(1+2\cos2\alpha)^2-a^2}{2\cdot4a\cos\alpha\cos2\alpha\cdot a(1+2\cos2\alpha)}=$$ $$=\frac{16\cos^2\alpha\cos^22\alpha+4\cos^22\alpha+4\cos2\alpha}{8\cos\alpha\cos2\alpha(1+2\cos2\alpha)}=\frac{4\cos^2\alpha\cos2\alpha+\cos2\alpha+1}{2\cos\alpha(1+2\cos2\alpha)}=$$ $$=\frac{4\cos^2\alpha\cos2\alpha+2\cos^2\alpha}{2\cos\alpha(1+2\cos2\alpha)}=\cos\alpha,$$ which says that indeed, $$\measuredangle BGA=2\alpha.$$

Answer 2

Solving without trigonometry. Just because.

If $CD||BE$, then $CE$ must be parallel to $BA$:

Since $\angle CBE =\alpha$ and $\angle DCF=\alpha$ then by the Corresponding Angles Postulate, then $CE||BA$.

$\angle DCF = 180^\circ-\angle DCE - \angle ECB $

$\angle DCF=180^\circ-\alpha-(180^\circ-2\alpha)=\alpha$

QED

Since we know that $CE||BA$. $$ \angle ECB = \angle EAB \qquad{\mathbb{it\ follows\ that:}}$$ $$180-2\alpha=\alpha+\frac{\pi}2$$

And because it is more fun and intuitive to think in degrees instead of radians, then $\alpha = 30^\circ$ $$\angle CBE = \angle CEB$$ $$\angle ECB+\angle CBE+\angle CEB = 180^\circ $$ $$\angle ECB = 180^\circ - 2\alpha$$ $$\angle ECB = 120^\circ$$

Also since we know that $CE||BA$ and $CB||EA$, then: $$\triangle CEB = \triangle EAB$$ $$\angle EAB = \angle ECB=120^\circ$$

And because $\angle EAB = \alpha +\frac{\pi}2 = 30^\circ+90^\circ$, it necessarily follows that: $$\angle DAB = 90^\circ$$ and because $\triangle CEB = \triangle EAB$, then $\angle ABE = 30^\circ$, which must mean:

$$\angle BGA = 90^\circ - \angle ABE = 90^\circ-30^\circ$$

$$\therefore \angle BGA = 60^\circ$$ This is a faster silver bullet for anyone with no expansive knowledge on trigonometry.