Question
In \( \triangle H \mathrm{HJ}, \mathrm{m} \angle H=(6 x-7)^{\circ}, \mathrm{m} \angle I=(x+10)^{\circ} \), and \( \mathrm{m} \angle J=(3 x-13)^{\circ} \). Find \( \mathrm{m} \angle H \).
Ask by Cox Mejia. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \mathrm{m} \angle H = 107^\circ \)
Solution
1. The sum of the angles in a triangle is \(180^\circ\). Therefore, we have:
\[
(6x-7) + (x+10) + (3x-13) = 180
\]
2. Combine like terms:
\[
6x + x + 3x -7 + 10 - 13 = 180 \quad \Longrightarrow \quad 10x - 10 = 180
\]
3. Solve for \(x\):
\[
10x = 180 + 10 \quad \Longrightarrow \quad 10x = 190 \quad \Longrightarrow \quad x = \frac{190}{10} = 19
\]
4. Now, substitute \(x = 19\) back into \(m \angle H = 6x-7\):
\[
m \angle H = 6(19)-7 = 114 - 7 = 107
\]
Thus, \(\mathrm{m} \angle H = 107^\circ\).
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Simplify this solution Beyond the Answer
To find the measures of the angles in triangle \( H \mathrm{HJ} \), we can use the fact that the sum of the angles in any triangle is \( 180^\circ \). Therefore, we have: \[ \mathrm{m} \angle H + \mathrm{m} \angle I + \mathrm{m} \angle J = 180^\circ \] Substituting the given expressions for the angles: \[ (6x - 7) + (x + 10) + (3x - 13) = 180 \] Combining like terms: \[ 6x + x + 3x - 7 + 10 - 13 = 180 \] \[ 10x - 10 = 180 \] Now, adding 10 to both sides: \[ 10x = 190 \] Dividing by 10: \[ x = 19 \] Now, we can find \( \mathrm{m} \angle H \): \[ \mathrm{m} \angle H = 6x - 7 = 6(19) - 7 = 114 - 7 = 107^\circ \] Thus, \( \mathrm{m} \angle H = 107^\circ \).
