In 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 three angles of the triangle?

Question

Sophia Alfred · Accepted Answer

#/_A = 73^circ,quad /_B = 34^circ, /_C = 73^circ#

In an isosceles triangle with #/_B# as the vertex, we have #/_A = /_C#.

Thus #/_B = 4x-2,qquad /_A = /_C = 8x+1#

Since #/_A+/_B+/_C = 180^circ#, we have

#4x-2+2(8x+1)=180^circ implies#

# 20x = 180^circ implies x = 9^circ#

Thus the angkes are

#/_A = 73^circ,quad /_B = 34^circ, /_C = 73^circ#

Violet Aldrich · Answer

undefined In an isosceles triangle, the angles opposite the congruent sides are congruent. Therefore, in triangle ABC, angles A and C are congruent. Let's denote the measure of angle B as $ 4x - 2 $ and the measure of angle A as $ 8x + 1 $.

Since the sum of angles in a triangle is $ 180^\circ $, we can write the equation:

$$ (4x - 2) + (8x + 1) + (4x - 2) = 180^\circ $$

Solve for $ x $:

$$ 4x - 2 + 8x + 1 + 4x - 2 = 180^\circ $$
$$ 16x - 3 = 180^\circ $$
$$ 16x = 183^\circ $$
$$ x = \frac{183^\circ}{16} $$

Now, find the measure of angle B:

$$ \text{Angle B} = 4x - 2 = 4 \left( \frac{183^\circ}{16} \right) - 2 $$
$$ \text{Angle B} = \frac{732^\circ}{16} - 2 $$
$$ \text{Angle B} = 45.75^\circ - 2 $$
$$ \text{Angle B} = 43.75^\circ $$

Now, find the measure of angle A (which is congruent to angle C):

$$ \text{Angle A} = 8x + 1 = 8 \left( \frac{183^\circ}{16} \right) + 1 $$
$$ \text{Angle A} = \frac{1464^\circ}{16} + 1 $$
$$ \text{Angle A} = 91.5^\circ + 1 $$
$$ \text{Angle A} = 92.5^\circ $$

Since angles A and C are congruent, angle C also measures $ 92.5^\circ $.

Therefore, the measures of all three angles of triangle ABC are:

Angle A: $ 92.5^\circ $

Angle B: $ 43.75^\circ $

Angle C: $ 92.5^\circ $