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The problem is as follows:

The figure from below shows a quadrilateral. Find the angle labeled $x$.

Sketch of the problem

The alternatives given in my book are as follows:

$\begin{array}{ll} 1.&8^{\circ}\\ 2.&10^{\circ}\\ 3.&15^{\circ}\\ 4.&18^{\circ}\\ 5.&5^{\circ}\\ \end{array}$

The only identity which I can recall in quadrilaterals is that the sum of the interior angles add up to $360^{\circ}$. I attempted several ways to split the given angles but I couldn't find something useful. Can someone help me with this?. How can this problem be solved with congruence or similarity or what?.

It would be helpful that an answer could include some sort of drawing or image because it would help to identify the way to find the angle.

Chris Steinbeck Bell
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1 Answers1

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Let $\angle ABD=\angle ADB=\theta$.

We have $\angle DAC=180^{\circ}-17x\Longrightarrow \angle DAB=180^{\circ}-12x\Longrightarrow \theta = 6x$

Therefore $\angle BDC=14x-6x=8x=\angle BCD\Longrightarrow \overline{BC}=\overline{BD}$

Since $\overline{AB}=\overline{AD}=\overline{BC}$, we have $\overline{AB}=\overline{AD}=\overline{BD}$, which implies that $\triangle ABD$ is an equilateral triangle.

Hence $\theta=60^{\circ}$, and since $\theta=6x$, we have $x=10^{\circ}$ as our final answer.enter image description here

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    We don't need $E$. Just observe that $\angle DAC = 180^\circ -17x$ and we have $\theta = 6x$. – player3236 Oct 22 '20 at 16:21
  • @player3236 Oh damn you're right! My bad. I shall edit my answer. –  Oct 22 '20 at 16:22
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    @Student1058, you're on roll today! +3! – cosmo5 Oct 22 '20 at 16:49
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    @cosmo5 Haha thanks. I'm in a really good mood today. –  Oct 22 '20 at 16:55
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    @Student1058 Other than the good mood I second that thanks to your patience on drawing with Geogebra it eased the delivery of the sketch. I think that the key here was to notice the existence of the equilateral triangle there. I ended up without noticing. Btw,I've recommend that you may use an increased thickness line for the equilateral triangle with a different color so it eases the understanding of your answer. – Chris Steinbeck Bell Oct 26 '20 at 02:52
  • I see. Thanks for the advice! –  Oct 26 '20 at 02:53