The image shows three triangles with different angle measures, and the task asks to determine the remaining angles of each triangle using the theorem that "In an isosceles triangle, the base angles are congruent." Let's solve each part:

Triangle a:

• The triangle is isosceles, as the two sides $AB$ and $AC$ are marked as equal.
• Angle $\angle B$ is given as $65^\circ$.
• Since the triangle is isosceles, $\angle A$ is the vertex angle, and the two base angles $\angle B$ and $\angle C$ are congruent.

To find $\angle A$:

• The sum of all angles in a triangle is $180^\circ$.
• Let $\angle B = \angle C = 65^\circ$.

Now: $\angle A + \angle B + \angle C = 180^\circ$ $\angle A + 65^\circ + 65^\circ = 180^\circ$ $\angle A = 180^\circ - 130^\circ = 50^\circ$

So, $\angle A = 50^\circ$.

Triangle b:

• This triangle is also isosceles, with sides $AB = AC$.
• The two base angles $\angle B$ and $\angle C$ are congruent, but no angle measure is given directly.

Since no angle is given, we'll label the base angles as $x$, and $\angle A$ as the vertex angle.

The equation is: $\angle A + 2x = 180^\circ$ Without more information, the specific values for $\angle A$ and the base angles can't be determined. Additional data is needed to solve this part.

Triangle c:

• This is not an isosceles triangle, and $\angle D$ is given as $130^\circ$.
• This appears to be part of a quadrilateral, so we might be able to calculate one of the triangle angles from the given information.

Since $\angle A$ and $\angle C$ form supplementary angles with $\angle D$ (since it's a straight line): $\angle A + 130^\circ = 180^\circ$ $\angle A = 180^\circ - 130^\circ = 50^\circ$

The remaining angles depend on whether additional information is available for the triangle.

Would you like more details on any of these solutions, or do you have specific questions about one of the triangles?

Here are some questions that might help expand this topic:

1. How do we identify isosceles triangles based on given side lengths?
2. Why are base angles congruent in isosceles triangles?
3. What is the general process to solve for angles in any triangle?
4. How can supplementary angles help in solving non-isosceles triangles?
5. How can we apply the angle sum property of triangles in different types of triangles?

Tip: In any triangle, the sum of the interior angles will always equal 180°, which is a helpful rule for solving for unknown angles.