In an isosceles triangle, the ratio between the vertical angle (angle at the vertex where the equal sides meet) and each base angle is 4 : 3. Find the measure of any one base angle in degrees.

Introduction / Context:
This aptitude problem involves angle relationships in an isosceles triangle. It tests your ability to use the given ratio between angles along with the fact that the sum of all interior angles of a triangle is 180 degrees. Recognising that an isosceles triangle has two equal base angles is essential for setting up the correct equation.

Given Data / Assumptions:

• We have an isosceles triangle.
• The vertical angle is the angle at the vertex where the two equal sides meet.
• The ratio of the vertical angle to a base angle is given as 4 : 3.
• There are two base angles and both are equal in measure.
• We must find the measure of one base angle in degrees.
• Sum of angles in any triangle is 180 degrees.

Concept / Approach:
Let the vertical angle be 4x degrees and each base angle be 3x degrees to reflect the given ratio 4 : 3. Because there are two base angles, the sum of the three interior angles of the triangle is:
4x + 3x + 3x = 180 degrees.
Solving this equation gives x and hence the measure of each base angle, 3x degrees. This straightforward algebraic approach is standard in triangle angle ratio problems.

Step-by-Step Solution:
Step 1: Let the vertical angle be 4x degrees. Step 2: Since the triangle is isosceles, both base angles are equal. Let each base angle be 3x degrees. Step 3: Use the fact that the sum of the angles in a triangle is 180 degrees. Step 4: Write the equation: 4x + 3x + 3x = 180. Step 5: Combine like terms: 4x + 3x + 3x = 10x. Step 6: So 10x = 180. Step 7: Solve for x: x = 180 / 10 = 18. Step 8: Each base angle is 3x degrees, so base angle = 3 * 18 = 54 degrees.

Verification / Alternative check:
Compute the vertical angle using x = 18 degrees: vertical angle = 4x = 4 * 18 = 72 degrees. Now check the sum of angles: 72 + 54 + 54 = 180 degrees, which matches the required total for a triangle. The ratio of vertical angle to base angle is 72 : 54, which simplifies by dividing both numbers by 18 to 4 : 3, confirming the ratio is correctly satisfied.

Why Other Options Are Wrong:
18 deg and 10 deg: These are too small for the base angle and would not satisfy both the 4 : 3 ratio and the 180 degree sum of angles.
72 deg: This is the measure of the vertical angle, not the base angle, in this configuration.
36 deg: Using 36 degrees as a base angle would not yield a vertical angle that maintains the 4 : 3 ratio with the base angles while also summing to 180 degrees.

Common Pitfalls:
Some students mistakenly assign 4x and 3x only once each and forget that there are two base angles. This leads to an equation 4x + 3x = 180 which is incorrect. Others may reverse the ratio and take the base angle as 4x and vertical as 3x, which conflicts with the wording of the question. Carefully reading the ratio description and remembering there are two equal base angles avoids these errors.

Final Answer:
The measure of each base angle is 54 degrees.