Question
In \( \triangle A B C, \mathrm{~m} \angle A=(2 x+6)^{\circ}, \mathrm{m} \angle B=(5 x-2)^{\circ} \), and \( \mathrm{m} \angle C=(3 x+6)^{\circ} \). Find \( \mathrm{m} \angle B \).
Ask by Dickson Allan. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \mathrm{m} \angle B = 83^\circ \).
Solution
1. Since \( \triangle ABC \) is a triangle, the sum of its interior angles is \( 180^\circ \):
\[
(2x+6)^\circ + (5x-2)^\circ + (3x+6)^\circ = 180^\circ.
\]
2. Combine like terms:
\[
2x + 5x + 3x + 6 - 2 + 6 = 180.
\]
This simplifies to:
\[
10x + 10 = 180.
\]
3. Solve for \( x \):
\[
10x = 180 - 10 = 170,
\]
\[
x = \frac{170}{10} = 17.
\]
4. Substitute \( x = 17 \) into the expression for \( \mathrm{m} \angle B \):
\[
\mathrm{m} \angle B = (5x - 2)^\circ = 5(17) - 2 = 85 - 2 = 83^\circ.
\]
Thus, \( \mathrm{m} \angle B \) is \( 83^\circ \).
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Simplify this solution Extra Insights
To find \( \mathrm{m} \angle B \) in triangle \( ABC \), we start by using the property that the sum of the interior angles of a triangle is \( 180^\circ \): \[ m \angle A + m \angle B + m \angle C = 180^\circ \] Substituting the given expressions for the angles, we get: \[ (2x + 6) + (5x - 2) + (3x + 6) = 180 \] Combining like terms: \[ 2x + 5x + 3x + 6 - 2 + 6 = 180 \] \[ 10x + 10 = 180 \] Now, we can solve for \( x \): \[ 10x = 180 - 10 \] \[ 10x = 170 \] \[ x = 17 \] Now we can find \( \mathrm{m}\angle B \): \[ m \angle B = 5x - 2 \] \[ m \angle B = 5(17) - 2 \] \[ m \angle B = 85 - 2 = 83^\circ \] Thus, \( \mathrm{m} \angle B = 83^\circ \).
