Question

if $overline{rs}congoverline{ru}$, $mangle rts = a + 1^{circ}$, and $mangle rtu = 2a$, what is the value of $a$?

Explanation:

Step1: Apply isosceles - triangle property

Since $\overline{RS}\cong\overline{RU}$ and $RT$ is a common side, and $\angle RST = \angle RUT=90^{\circ}$, triangles $\triangle RST$ and $\triangle RUT$ are congruent by the Hypotenuse - Leg (HL) congruence criterion. So, $\angle RTS=\angle RTU$.

Step2: Set up the equation

We know that $m\angle RTS=a + 1^{\circ}$ and $m\angle RTU = 2a$. Set $a + 1=2a$.

Step3: Solve the equation for $a$

Subtract $a$ from both sides of the equation $a + 1=2a$. We get $1=2a - a$, so $a = 1$.

Answer:

$1$