Question

solving for angle measures of isosceles triangles
the vertex angle of an isosceles triangle measures ( 42 ^ { circ } ). a base angle in the triangle has a measure given by ( ( 2 x + 3 ) ^ { circ } ). what is the value of ( x )? what is the measure of each base angle?
( x = )
each base angle measures ( )°.

Explanation:

Step1: Use the angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). In an isosceles triangle, the two base angles are equal. Let the vertex angle \(A = 42^{\circ}\) and each base angle \(B = C=(2x + 3)^{\circ}\). Then \(A + B + C=180^{\circ}\), so \(42+(2x + 3)+(2x + 3)=180\).

Step2: Simplify the equation

Combine like terms: \(42+4x + 6=180\), which simplifies to \(4x+48 = 180\).

Step3: Solve for \(x\)

Subtract \(48\) from both sides: \(4x=180 - 48\), so \(4x=132\). Then divide both sides by \(4\): \(x=\frac{132}{4}=33\).

Step4: Find the measure of the base angle

Substitute \(x = 33\) into \((2x + 3)\). We get \(2\times33+3=66 + 3=69\).

Answer:

\(x = 33\)
Each base angle measures \(69^{\circ}\)