Angle X Circle D1B366

1. **State the problem:** We are given a circle with center $O$, chords $AH$ and $BC$ which are parallel, and segments $OG$ and $OH$ of equal length. We need to find the size of angle $x$ at point $B$ formed by lines $BA$ and $BE$. 2. **Identify key facts and formulas:** - Since $O$ is the center, $OG$ and $OH$ being equal means triangle $OGH$ is isosceles. - $AH \parallel BC$ implies alternate interior angles are equal. - The angle at $O$ between $OH$ and $OD$ is $70^\circ$. - The angle $x$ at $B$ is between $BA$ and $BE$. 3. **Analyze the isosceles triangle $OGH$:** Since $OG = OH$, triangle $OGH$ is isosceles with $OG = OH$. 4. **Use the given $70^\circ$ angle at $O$:** The angle between $OH$ and $OD$ is $70^\circ$. Since $OG$ and $OH$ are equal, angles at $G$ and $H$ related to $O$ are equal. 5. **Use parallel lines $AH \parallel BC$:** Alternate interior angles formed by transversal $AB$ imply that angle $x$ at $B$ equals the angle between $AH$ and $AB$. 6. **Calculate angle $x$:** Because $OG = OH$, triangle $OGH$ is isosceles with vertex angle $70^\circ$ at $O$. The base angles at $G$ and $H$ are each: $$\frac{180^\circ - 70^\circ}{2} = \frac{110^\circ}{2} = 55^\circ$$ Since $AH \parallel BC$, angle $x$ at $B$ equals $55^\circ$ by alternate interior angles. **Final answer:** $$x = 55^\circ$$