Final answer:
The question is a problem about finding the value of x and the measures of angles in an isosceles triangle. By setting the base angles equal to each other and solving for x, we find that x equals 5. Then, we use the triangle angle sum theorem to determine that each base angle measures 47 degrees and the vertex angle measures 86 degrees.
Step-by-step explanation:
In the triangle ARST, if RT = ST, then triangle RST is an isosceles triangle with the base angles R and S being equal. The triangle angle sum theorem states that the angles in a triangle add up to 180 degrees.
We have the equations from the given information:
m∠R = 9x + 2,
m∠S = 13x – 18, and
m∠T = 17x + 1.
Since RT = ST, the base angles (m∠R and m∠S) are equal, so:
9x + 2 = 13x – 18.
By solving this equation for x, we have:
4x = 20,
x = 5.
The measure of angle T, which is not one of the base angles, is then calculated using the sum of the angles in the triangle:
m∠T = 180 – (m∠R + m∠S),
m∠T = 180 – ((9x + 2) + (9x + 2)),
m∠T = 180 – (18x + 4),
m∠T = 180 – (18 * 5 + 4),
m∠T = 180 – 94,
m∠T = 86 degrees.
Knowing x, you can also find the measures of angles R and S, each being 9x + 2 or 9 * 5 + 2, which is 47 degrees.