Question

suppose that \triangle klm is isosceles with base \overline{kl}.
suppose also that m\angle k=(3x + 23)^{circ} and m\angle l=(4x + 12)^{circ}.
find the degree measure of each angle in the triangle.

Explanation:

Step1: Use the property of isosceles triangle

Since \(\triangle KLM\) is isosceles with base \(\overline{KL}\), then \(\angle K=\angle L\). So, \(3x + 23=4x+12\).
Solve for \(x\):
\(3x+23 = 4x + 12\)
\(23-12=4x - 3x\)
\(x = 11\)

Step2: Calculate \(m\angle K\) and \(m\angle L\)

Substitute \(x = 11\) into the expressions for \(\angle K\) and \(\angle L\).
For \(m\angle K=(3x + 23)^{\circ}\), \(m\angle K=(3\times11 + 23)^{\circ}=(33+23)^{\circ}=56^{\circ}\)
For \(m\angle L=(4x + 12)^{\circ}\), \(m\angle L=(4\times11+12)^{\circ}=(44 + 12)^{\circ}=56^{\circ}\)

Step3: Calculate \(m\angle M\)

Use the triangle - angle sum theorem (\(m\angle K+m\angle L+m\angle M = 180^{\circ}\))
\(m\angle M=180-(m\angle K + m\angle L)\)
Since \(m\angle K=m\angle L = 56^{\circ}\), then \(m\angle M=180-(56 + 56)=180 - 112=68^{\circ}\)

Answer:

\(m\angle K = 56^{\circ}\), \(m\angle L=56^{\circ}\), \(m\angle M = 68^{\circ}\)