Question

what is the value of x in the figure? show calculator

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. In triangle ABC, we know two angles: one is 80° and the other is 57°.

Step2: Calculate the third - angle of triangle ABC

Let the third - angle of triangle ABC be \(y\). Then \(y=180-(80 + 57)\).
\[y=180 - 137=43^{\circ}\]

Step3: Consider the relationship between angles in the figure

In the figure, \(x\) and \(y\) are related such that \(x\) and \(y\) are the non - overlapping parts of the angle at B. Since \(AB = BC\) and \(AD = CD\), the diagonal \(BD\) is a line of symmetry. The angle at B in triangle ABC and the angle at B in triangle BCD are symmetric. We know that the sum of angles around a point is 360°. In this case, considering the angles at B in the two triangles formed by the diagonal, we can also use the fact that in triangle ABC, we found the third - angle \(y = 43^{\circ}\). And since the figure has some symmetry properties (isosceles - like triangles formed by the diagonal), we can find \(x\) using the angle - sum property of a triangle in the other triangle formed by the diagonal. Another way is to note that in triangle ABC, we have:
\[x=43^{\circ}\]

Answer:

43