Question

fill in the sentence stem to complete the proof for \\( \overline{x z} \cong \overline{w z} \\) use the word bank to help you fill the stem in. you will use the same word more than once. \\( \overleftrightarrow{x z} \\) and \\( \overleftrightarrow{w z} \\) are tangent to circle o. line xz and line wz are ____ to circle o you can connect ox to make a __ you can connect ow to make a __ both of those line segments are __ you can connect oz to make a __ of the two triangles both triangles that you created are __, so all of the parts, legs and angles are also ____. word bank: congruent radius leg chord congruent tangent hypotenuse radius congruent diameter

Explanation:

• Given that \( \overleftrightarrow{XZ} \) and \( \overleftrightarrow{WZ} \) are tangent to the circle, so the first blank is "tangent".
• \( OX \) and \( OW \) are radii of the circle (a line segment from the center of a circle to a point on the circle).
• All radii of a circle are congruent, so \( OX\cong OW \).
• \( OZ \) is the hypotenuse of right - triangles \( \triangle OXZ\) and \( \triangle OWZ\) (by the property that a tangent to a circle is perpendicular to the radius at the point of tangency, so \( \angle OXZ=\angle OWZ = 90^{\circ}\)).
• By the Hypotenuse - Leg (HL) congruence criterion for right - triangles (\( OZ = OZ\) (common hypotenuse) and \( OX=OW\) (radii/congruent)), \( \triangle OXZ\cong\triangle OWZ\). If two triangles are congruent, then all their corresponding parts (legs, angles) are congruent.

Answer:

Line \( XZ \) and Line \( WZ \) are \(\text{tangent}\) to circle \( O\).
You can connect \( OX \) to make a \(\text{radius}\).
You can connect \( OW \) to make a \(\text{radius}\).
Both of those line segments are \(\text{congruent}\).
You can connect \( OZ \) to make a \(\text{hypotenuse}\) of the two triangles.
Both triangles that you created are \(\text{congruent}\), so all of the parts, legs and angles are also \(\text{congruent}\).