Question

if triangle rst is isosceles, solve for x. (15x - 31)° (9x + 11)° o 42 degrees o 74 degrees o 158 degrees o 32 degrees

Explanation:

Step1: Identify equal - angles

Since the triangle is isosceles, $\angle R=\angle S$. Let's set up an equation using the angle - sum property of a triangle. The sum of the interior angles of a triangle is $180^{\circ}$. So, $(15x - 31)+(15x - 31)+(9x + 11)=180$.

Step2: Combine like - terms

First, combine the $x$ terms: $15x+15x + 9x=39x$, and combine the constant terms: $-31-31 + 11=-51$. The equation becomes $39x-51 = 180$.

Step3: Solve for $x$

Add 51 to both sides of the equation: $39x=180 + 51=231$. Then, divide both sides by 39: $x=\frac{231}{39}=\frac{77}{13}\approx5.92$. But we can also use the fact that in an isosceles triangle, if we assume the non - equal angle is $\angle T$. Then, $15x-31=9x + 11$.

Step4: Solve the new equation for $x$

Subtract $9x$ from both sides: $15x-9x-31=9x-9x + 11$, which gives $6x-31 = 11$. Add 31 to both sides: $6x=11 + 31=42$. Divide both sides by 6: $x = 7$.

Step5: Find the measure of $\angle T$

Substitute $x = 7$ into the expression for $\angle T$: $\angle T=(9x + 11)^{\circ}$. So, $\angle T=9\times7 + 11=63+11=74^{\circ}$.

Answer:

74 degrees