In a triangle ABC, sides AB and AC are of equal length. If `angleB=35^(@)`, find the measure of `angleA and angleC`.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the triangle
We are given that triangle ABC has sides AB and AC of equal length. This means that triangle ABC is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are also equal.

### Step 2: Identify the given information
- Angle B = 35 degrees
- Sides AB = AC (equal sides)

### Step 3: Use the property of isosceles triangles
Since AB = AC, the angles opposite these sides must be equal. Therefore:
- Angle C = Angle B = 35 degrees

### Step 4: Use the angle sum property of triangles
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Substituting the known values:
\[ \text{Angle A} + 35^\circ + 35^\circ = 180^\circ \]

### Step 5: Simplify the equation
Combine the angles:
\[ \text{Angle A} + 70^\circ = 180^\circ \]

### Step 6: Solve for Angle A
To find Angle A, subtract 70 degrees from both sides:
\[ \text{Angle A} = 180^\circ - 70^\circ \]
\[ \text{Angle A} = 110^\circ \]

### Conclusion
Thus, the measures of the angles in triangle ABC are:
- Angle A = 110 degrees
- Angle B = 35 degrees
- Angle C = 35 degrees