Question

1. the ratio of the measure of the vertex angle to the base angle of an isosceles triangle is 3:5. find the measure of the vertex angle.

Explanation:

Step1: Define angle variables

Let the vertex angle be $8x$, each base angle be $5x$.

Step2: Sum angles of triangle

Triangle angle sum is $180^\circ$.
$$8x + 5x + 5x = 180^\circ$$

Step3: Solve for $x$

Simplify and calculate $x$.
$$18x = 180^\circ \implies x = 10^\circ$$

Step4: Calculate vertex angle

Substitute $x$ into vertex angle expression.
$$8x = 8 \times 10^\circ = 80^\circ$$

Answer:

$80^\circ$