Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
#### Given Information:
1. \(\overline{WY} \cong \overline{XZ}\)
2. \(\overline{WZ} \perp \overline{ZY}\)
3. \(\overline{WZ} \perp \overline{ZY}\)
#### Explanation
To solve the problem, we need to explain why triangles \(\triangle WZY\) and \(\triangle XYZ\) are congruent.
From the given information:
1. The segments \(\overline{WY}\) and \(\overline{XZ}\) are congruent.
2. The segment \(\overline{WZ}\) is perpendicular to segment \(\overline{ZY}\).
These congruent segments and right angles play a vital role in determining the congruence of the triangles.
#### Analysis of the Diagram
The provided diagram shapes a quadrilateral \(WXYZ\) with:
- An intersection at segments \(\overline{WZ}\) and \(\overline{ZY}\).
- The criteria that make segments equal and angles perpendicular allow us to apply triangle congruence principles:
* Side-Side-Side (SSS) congruence
* Side-Angle-Side (SAS) congruence
- Each triangle shares a pair of sides \(\overline{WY}\) and \(\overline{XZ}\) and includes the perpendicular angle shared by \(\overline{WZ}\) making them perfect right triangles.
#### Conclusion
By the properties established:
- Given that \(\overline{WY} \cong \overline{XZ}\),
- Both include right angles at \(\overline{WZ}\) and \(\overline{ZY}\), and as they share identical criteria:
We can conclude:
\[
\triangle WZY \cong \triangle XYZ
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