6) Find the missing angle.

Solution:

In the given triangle, the side AB is equal to side BC (indicated by tick marks). This means the triangle is isosceles.

In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, \(\angle BAC = \angle BCA\).

We are given that \(\angle BCA = 50^{\circ}\).

So, \(\angle BAC = 50^{\circ}\).

The sum of angles in a triangle is \(180^{\circ}\).

\(\angle ABC = 180^{\circ} - (\angle BAC + \angle BCA)\)

\(\angle ABC = 180^{\circ} - (50^{\circ} + 50^{\circ})\)

\(\angle ABC = 180^{\circ} - 100^{\circ} = 80^{\circ}\).

The question asks for the angle marked with '?'. This angle is \(\angle BDA\). In triangle ABD, we have \(\angle BAD = 50^{\circ}\) and \(\angle ABD\) is part of \(\angle ABC\). However, the diagram shows point D on the side AC. Let's assume the question is asking for \(\angle ADB\) in triangle ABD.

We need more information to find \(\angle ADB\) directly from the given values if D is on AC. But looking at the diagram, D appears to be a point on AC and the angle marked with '?' is \(\angle ADB\).

Let's reconsider the diagram. It is likely that the '?' refers to \(\angle ABD\) or \(\angle ADB\) within the triangle ABD.

If we assume that the triangle ABC is isosceles with AB=BC and \(\angle BCA = 50^{\circ}\), then \(\angle BAC = 50^{\circ}\) and \(\angle ABC = 80^{\circ}\).

However, there is a point D and an angle marked '?'. If D is on AC, and the angle '?' is \(\angle ADB\), we cannot solve it with the given information.

Let's assume the question intends to ask for \(\angle ADB\) within triangle ABD, where D is a point on AC. If D is a vertex of another triangle, we lack information.

Let's assume the '?' refers to an angle within triangle ABD. We know \(\angle BAD = 50^{\circ}\). We need \(\angle ABD\) or \(\angle ADB\).

Let's examine the possibility that D is a point on BC or AC. The label D is near the segment AC.

If we assume the question is asking for \(\angle ADB\) and D is on AC, we cannot solve it.

Let's consider the possibility that the '?' is \(\angle ADB\) and the triangle is ABD. We know \(\angle DAB = 50^{\circ}\). We need more information.

Let's look at the labels A, B, C, D. It seems ABC is a triangle and D is a point on AC.

If the question is asking to find the angle \(\angle ADB\) where D is on AC:

We know \(\angle BAC = 50^{\circ}\), \(\angle BCA = 50^{\circ}\), \(\angle ABC = 80^{\circ}\).

Without knowing the position of D on AC, or another angle in triangle ABD or triangle BDC, we cannot find \(\angle ADB\).

Let's consider another interpretation. Perhaps the '?' refers to \(\angle ABC\), and the 80 degrees is \(\angle ADB\) or similar. But that contradicts the labels.

Let's re-examine the image carefully. The '?' is placed at vertex D of a triangle that seems to be ABD. If D is on AC, then angle ABC is 80 degrees. Angle BAC is 50 degrees. Angle BCA is 50 degrees. If D is a point on AC, then we need more info.

However, if we consider triangle ABD, we know \(\angle BAD = 50^{\circ}\). We don't know \(\angle ABD\) or \(\angle ADB\).

Let's consider the possibility that D is not on AC, but is a separate vertex. Then we have triangle ABD and triangle BCD.

If the question is asking for \(\angle BDC\) or \(\angle ADB\):

Let's assume the label 'D' is a vertex and the angle marked '?' is \(\angle ADB\). We are given \(\angle BAC = 50^{\circ}\), \(\angle BCA = 50^{\circ}\), \(\angle ABC = 80^{\circ}\). This forms triangle ABC.

If D is a vertex, and the diagram is meant to be two triangles ABC and ABD.

Let's assume D is a point on AC, and the '?' is the angle \(\angle ADB\). We have \(\angle BAC = 50^{\circ}\).

If we consider the case where D is a point on AC, and we are asked to find \(\angle ADB\).

Let's assume the diagram is split into two triangles by a line segment BD.

If the '?' refers to an angle in triangle ABD, then we know \(\angle BAD = 50^{\circ}\). We need more information about \(\angle ABD\) or \(\angle ADB\).

Let's look at the other numbers. Number 6 is labeling this diagram.

Let's assume the question intends for triangle ABC to be isosceles with AB=BC, \(\angle BAC = \angle BCA = 50^{\circ}\) and \(\angle ABC = 80^{\circ}\). And point D is on AC. Then the angle marked '?' is \(\angle ADB\). Without knowing where D is on AC, we cannot determine \(\angle ADB\).

Let's reconsider the diagram and the possibility that D is a vertex of triangle ABD, and maybe BD is an altitude or angle bisector or median. There is no indication of this.

Let's assume there's a typo in the question or diagram.

If we assume D is a point on AC and BD is the angle bisector of \(\angle ABC\), then \(\angle ABD = \angle CBD = 40^{\circ}\). In triangle ABD, \(\angle BAD = 50^{\circ}\), \(\angle ABD = 40^{\circ}\). Then \(\angle ADB = 180^{\circ} - (50^{\circ} + 40^{\circ}) = 180^{\circ} - 90^{\circ} = 90^{\circ}\).

If we assume BD is the altitude from B to AC, then \(\angle BDA = 90^{\circ}\). In triangle ABD, \(\angle BAD = 50^{\circ}\), so \(\angle ABD = 180^{\circ} - 90^{\circ} - 50^{\circ} = 40^{\circ}\). This is consistent with \(\angle ABC = 80^{\circ}\) if D is the foot of the altitude and the triangle is isosceles.

Let's assume BD is the median to AC. Then D is the midpoint of AC. We still need more info.

Given the simplicity of other problems, it is likely that BD is the altitude or angle bisector.

Let's assume BD is the altitude, so \(\angle BDA = 90^{\circ}\).

Then in triangle ABD, \(\angle BAD = 50^{\circ}\), \(\angle BDA = 90^{\circ}\), so \(\angle ABD = 180^{\circ} - 90^{\circ} - 50^{\circ} = 40^{\circ}\). This implies \(\angle ABC = 80^{\circ}\) if D is on AC. This is consistent with ABC being isosceles.

So, if we assume BD is the altitude to AC, then the angle at D is 90 degrees. However, the '?' is not at D, but within triangle ABD.

Let's assume the '?' refers to the angle \(\angle ADB\). If BD is the altitude, then \(\angle BDA = 90^{\circ}\). The '?' is marking this angle.

Answer: 90° (assuming BD is the altitude to AC).