Chapter 9.4, Problem 16WE

To determine

To prove:In the same circle or in congruent circles, congruent chords have congruent arcs.

Answer to Problem 16WE

Below proof shows that in a circle, congruent chords have congruent arcs.

Explanation of Solution

Giveninformation: In below given figure,a circle is given in whichchords $RS$ and $TU$ are congruent.

  

Concept used:Use SSS congruency that says thatif all corresponding sides are congruent in both the triangles. Also use the fact that in a circle, if two angles on center are congruent, their corresponding arcs are also congruent.

Proof:In given figure, draw radii $\overline{OR},\overline{OS},\overline{OT}$ and $\overline{OU}$ ,so $\overline{OR}\cong \overline{OT}$ and $\overline{OS}\cong \overline{OU}$ because they are all radii of the same circle. Also as $\overline{RS}\cong \overline{TU}$ . Then $\Delta ROS\cong \Delta TOU$ by SAS and corresponding angles 1 and 2 will be congruent, being corresponding parts of both these triangles.

Conclusion: Thus,it proves that as angle 1 and 2 are congruent, their corresponding arcs $RS$ and $TU$ will also be congruent to each other.