Question

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what is the measure of the vertex angle in the given isosceles triangle?
measure of the vertex angle =

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle, the base - angles are equal. Let the vertex angle be \(A=(x + 10)^{\circ}\) and the base - angles be \(B = C=(2x + 5)^{\circ}\).
By the angle - sum property of a triangle \(A + B + C=180^{\circ}\).
Substitute the values: \((x + 10)+(2x + 5)+(2x + 5)=180\).

Step2: Simplify the equation

Combine like terms: \(x+10 + 2x+5+2x + 5=180\).
\(5x+20 = 180\).
Subtract 20 from both sides: \(5x=180 - 20\).
\(5x=160\).
Divide both sides by 5: \(x=\frac{160}{5}=32\).

Step3: Find the vertex angle

Substitute \(x = 32\) into the vertex - angle formula \(A=(x + 10)^{\circ}\).
\(A=(32+10)^{\circ}\).

Answer:

\(42\)