# How to find the angle in between two triangles?

The problem is indicated in the figure from below:

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&10^{\circ}\\ 2.&15^{\circ}\\ 3.&20^{\circ}\\ 4.&18^{\circ}\\ \end{array}$$

What exactly should be done here?. I'm stuck. The only thing which I can spot is that:

$$\angle DAC + x = 55$$

$$\angle BAC + 40 = \angle BCD + 55$$

But without any other further knowledge, I don't know what else can be done here?. Can someone help me here?. Can this problem be solved relying only in euclidean geometry?.

The other thing which I can spot is since it says $$BD \parallel AC$$

This means:

$$\angle BDA = \angle DAC$$

But again this information doesn't really help me much into solving this problem. Can someone guide me further?. Please include a drawing in the answer because I'm lost. Does this requires a construction?.

• Hint: You can draw a circle centered at B, passing through A, C, D, and then use the inscribed angle theorem
– user632577
Jan 6, 2021 at 23:47

If $$AB=BC=BD$$, then $$B$$ is the centre of circle with $$A,C,D$$ on the circumference and the angle $$\angle ADC$$ is half the angle $$\angle ABC$$, i.e. $$20^\circ$$

• Note that this solution works even without assuming BD // AC. Jan 7, 2021 at 9:16
• @FedericoPoloni Yep, I didn't need that at all - I saw it but I didn't even consider it after seeing $B$ as the circle origin. Jan 7, 2021 at 15:05

At first (blue angles), notice how the $$ABC$$ triangle is isosceles (since $$AB=BC$$), and you get that the bottom angles are both $$70$$ (because $$\frac{180-40}{2}=70$$). Then, since $$BD\| AC$$, you can find that the angle next to the $$40$$º is also $$70$$, and that the big angle that contains the $$x$$ is $$55$$.

Finally (red angles), we also know $$AB=BD$$, so the triangle $$ABD$$ is also isosceles and we know one angle is $$40+70=110$$, so the other two must be $$35$$ (because $$\frac{180-110}{2}=35$$).

So your $$x$$ is $$\boxed{x=55-35=20}$$

Ask me is there's anything you don't understand, maybe I explained it too fast.

• You explained it perfectly well. Jan 10, 2021 at 5:09