Question

suppose that ( \triangle ghi ) is isosceles with base ( overline{ih} ). suppose also that ( mangle g=(3x + 53)^{circ} ) and ( mangle i=(2x + 25)^{circ} ). find the degree measure of each angle in the triangle.

Explanation:

Step1: Use the property of isosceles triangle

Since \(\triangle GHI\) is isosceles with base \(\overline{IH}\), then \(\angle H=\angle I=(2x + 25)^{\circ}\).
By the angle - sum property of a triangle (\(m\angle G+m\angle H+m\angle I = 180^{\circ}\)), substitute the angle measures:
\((3x + 53)+(2x + 25)+(2x + 25)=180\).

Step2: Simplify the equation

Combine like terms:
\(3x+53+2x + 25+2x + 25=180\)
\(7x+103 = 180\).
Subtract 103 from both sides:
\(7x=180 - 103\)
\(7x=77\).

Step3: Solve for \(x\)

Divide both sides by 7:
\(x=\frac{77}{7}=11\).

Step4: Find \(m\angle G\)

Substitute \(x = 11\) into \(m\angle G=(3x + 53)^{\circ}\):
\(m\angle G=3\times11+53=33 + 53=86^{\circ}\).

Step5: Find \(m\angle H\) and \(m\angle I\)

Substitute \(x = 11\) into \(m\angle I=(2x + 25)^{\circ}\) (and \(m\angle H=m\angle I\)):
\(m\angle I=2\times11+25=22 + 25=47^{\circ}\), \(m\angle H = 47^{\circ}\).

Answer:

\(m\angle G = 86^{\circ}\), \(m\angle H=47^{\circ}\), \(m\angle I = 47^{\circ}\)