We are given three geometry problems involving circles, arcs, and chords. 1. Find $x$ in the first figure, where the inscribed angle intercepts an arc of $77^\circ$.

1. Problem Description

We are given three geometry problems involving circles, arcs, and chords.

1. Find $x$ in the first figure, where the inscribed angle intercepts an arc of $77^\circ$.

2. Find $x$ in the second figure, where two sides of a triangle formed by radii are congruent and the measure of one angle is $68^\circ$.

3. Find $x$ in the third figure, where chords $AB$ and $CB$ are congruent, and the arcs intercepted by the given central angles have measures $118^\circ$ and $9x-6^\circ$ respectively.

2. Solution Steps

1. In the first problem, the inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Therefore, the measure of the arc $JK$ is twice the measure of the inscribed angle formed by the chord intercepted by the arc $LM$.

$x = 2(77^\circ) = 154^\circ$

2. In the second problem, we are given that $SR = ST$. This means that $\triangle SRT$ is an isosceles triangle. Therefore, $\angle SRT = \angle STR = 68^\circ$. The angles in a triangle sum to $180^\circ$.

$\angle RST + \angle SRT + \angle STR = 180^\circ$

$x + 68 + 68 = 180$

$x + 136 = 180$

$x = 180 - 136$

$x = 44^\circ$

3. In the third problem, since chords $AB$ and $CB$ are congruent, the arcs they intercept are also congruent. Thus, the arc $AB$ has the same degree measure as the arc $CB$, $118^\circ = 9x-6$. Therefore:

$9x - 6 = 118$

$9x = 118 + 6$

$9x = 124$

$x = \frac{124}{9}$

3. Final Answer

1. $x = 154$

2. $x = 44$

3. $x = \frac{124}{9}$