Question

1. the vertex angle of an isosceles triangle is 42 more than 3 times the base angles. find the vertex angle. the vertex angle is 82.8°. the vertex angle is 40.8°. the vertex angle is 27.6°. the vertex angle is 124.8°.

Explanation:

Step1: Define Variables

Let the measure of each base angle be \( x \) degrees. Then the vertex angle is \( 3x + 42 \) degrees (since it's 42 more than 3 times the base angle). In an isosceles triangle, the sum of the interior angles is \( 180^\circ \), and there are two equal base angles. So the equation is \( 2x+(3x + 42)=180 \).

Step2: Solve the Equation

Combine like terms: \( 2x+3x + 42=180 \) simplifies to \( 5x+42 = 180 \). Subtract 42 from both sides: \( 5x=180 - 42=138 \). Then divide both sides by 5: \( x=\frac{138}{5}=27.6 \) degrees (this is the base angle).

Step3: Find Vertex Angle

Now, substitute \( x = 27.6 \) into the expression for the vertex angle: \( 3x+42=3\times27.6 + 42=82.8+42 = 124.8 \) degrees.

Answer:

The vertex angle is \( 124.8^\circ \). (Corresponding to the option: The vertex angle is \( 124.8^\circ \).)