Question

In an isosceles triangle ABC, B is the vertex. The measure of angle B can be represented as $(4x - 2)$. The measure of angle A can be represented as $(8x + 1)$. Find the measure of all the three angles of the triangle?

Answer 1

Hint: An isosceles triangle is a triangle with two sides being equal. The vertex of the triangle is nothing but the vertex which is common to the two equal sides. Sum of angles in a triangle (be it equilateral, isosceles or scalene) is 180°. Angles opposite to equal sides are equal which means that the two angles other than the vertex angle (also known as base angles) are equal in an isosceles triangle.

Complete step-by-step answer:
First, let’s draw a diagram which shows all the information in the question given above.

ABC is an isosceles triangle with vertex B
The sides marked – AB and BC are the two equal sides and hence the angles opposite to them are also equal.
$\left| \!{\underline {\,
  A \,}} \right. = \left| \!{\underline {\,
  C \,}} \right. = 8x + 1$
We know that the sum of angles in a triangle is 180°.
Therefore,
$\left| \!{\underline {\,
  A \,}} \right. + \left| \!{\underline {\,
  B \,}} \right. + \left| \!{\underline {\,
  C \,}} \right. = 180^\circ $
$(4x - 2) + (8x + 1) + (8x + 1) = 180^\circ $
$20x = 180^\circ $
$x = 9^\circ $
Now substituting the value of x, we will get the values of all three angles.
$\left| \!{\underline {\,
  A \,}} \right. = 8x + 1 = 8(9) + 1 = 73^\circ $
$\left| \!{\underline {\,
  B \,}} \right. = 4x - 2 = 4(9) - 2 = 34^\circ $
$\left| \!{\underline {\,
  C \,}} \right. = 8x + 1 = 8(9) + 1 = 73^\circ $
Hence, the angles of the triangle are 73°,34°,73°

Note: Most of the students make a very common mistake by assuming that the given angles are the values of the two equal angles. One should be careful while reading the question and carefully draw the diagram because a simple mistake can give a completely wrong answer. We need to remember the basic properties of triangles like base angles in an isosceles triangle are equal and sum of angles in a triangle is 180°.