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 = square )
each base angle measures ( square )

Explanation:

Step1: Use the angle - sum property of a triangle

In an isosceles triangle, the base angles are equal. The sum of the interior angles of a triangle is \(180^{\circ}\). Let the vertex angle \(V = 42^{\circ}\) and each base angle \(B=(2x + 3)^{\circ}\). Then \(V+2B=180^{\circ}\).
Substitute \(V = 42^{\circ}\) and \(B=(2x + 3)^{\circ}\) into the equation: \(42+2(2x + 3)=180\).

Step2: Simplify the equation

First, expand \(2(2x + 3)\) using the distributive property \(a(b + c)=ab+ac\). So \(2(2x + 3)=4x+6\).
The equation becomes \(42+4x + 6=180\). Combine like terms: \(4x+(42 + 6)=180\), i.e., \(4x+48=180\).
Subtract 48 from both sides: \(4x=180 - 48\). So \(4x=132\).

Step3: Solve for \(x\)

Divide both sides of the equation \(4x = 132\) by 4. Using the formula \(x=\frac{132}{4}\), we get \(x = 33\).

Step4: Find the measure of the base angle

Substitute \(x = 33\) into the formula for the base angle \(B=(2x + 3)^{\circ}\).
\(B=(2\times33+3)^{\circ}=(66 + 3)^{\circ}=69^{\circ}\).

Answer:

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