18. (58) Line $$ x $$ is tangent to $$ \odot C $$ at $$ G $$, and line $$ y $$ passes through $$ C $$. Lines $$ x $$ and $$ y $$ intersect at $$ H $$. If $$ m \angle C H G=52^{\circ} $$, determine $$ m \angle H C G $$.

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To determine $$ m \angle H C G $$, let's analyze the given information step by step.

1. **Understanding the Diagram:**
   - **Circle $$ C $$:** $$ C $$ is the center of the circle.
   - **Point $$ G $$:** $$ G $$ lies on the circle, and line $$ x $$ is tangent to the circle at $$ G $$.
   - **Tangent Line $$ x $$:** Since $$ x $$ is tangent at $$ G $$, it is perpendicular to the radius $$ C G $$. Therefore, $$ x \perp C G $$, making $$ \angle C G x = 90^\circ $$.
   - **Line $$ y $$:** Passes through $$ C $$ and intersects $$ x $$ at point $$ H $$.

2. **Analyzing Triangle $$ C H G $$:**
   - **Given Angle:** $$ \angle C H G = 52^\circ $$.
   - **Perpendicular Relationship:** Since $$ x $$ is perpendicular to $$ C G $$, $$ \angle C G H = 90^\circ $$.

3. **Calculating the Remaining Angle:**
   - The sum of angles in triangle $$ C H G $$ is $$ 180^\circ $$.
   $$
   \angle H C G + \angle C H G + \angle C G H = 180^\circ
   $$
   $$
   \angle H C G + 52^\circ + 90^\circ = 180^\circ
   $$
   $$
   \angle H C G = 180^\circ - 52^\circ - 90^\circ
   $$
   $$
   \angle H C G = 38^\circ
   $$

**Final Answer:** $$ m \angle H C G = 38^\circ $$
$$ m \angle H C G = 38^\circ $$

(58) Line is tangent to at , and
line passes through . Lines and
intersect at . If ,
determine .

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(58) Line is tangent to at , and line passes through . Lines and intersect at . If , determine .