Question

in \\( \triangle g h i, mathrm{m} angle g=(x + 16)^{circ}, mathrm{m} angle h=(x + 9)^{circ}, \\) and \\( mathrm{m} angle i=(7 x + 2)^{circ} \\). find \\( mathrm{m} angle g \\).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle G+m\angle H + m\angle I=180^{\circ}\).
Substitute the given angle expressions: \((x + 16)+(x + 9)+(7x+2)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((x+x + 7x)+(16 + 9+2)=180\), which is \(9x+27 = 180\).

Step3: Solve for \(x\)

Subtract 27 from both sides: \(9x=180 - 27\), so \(9x=153\).
Divide both sides by 9: \(x=\frac{153}{9}=17\).

Step4: Find \(m\angle G\)

Since \(m\angle G=(x + 16)^{\circ}\), substitute \(x = 17\) into the expression. Then \(m\angle G=(17+16)^{\circ}\).

Answer:

\(33^{\circ}\)