Arc and Chord Properties:

**Theorem 14:** *In equal circles, if two arcs subtends equal angles at the center, then the arcs are equal.*

Given: Two circles with centers respectively.

Two subtend equal angles at the respective centers.

To Prove:

Proof: Join

Now consider

(radius of the same circle)

(radius of the same circle)

Now since the two circles are equal, it means that their radius are equal.

Hence

Also given that

Hence (S.A.S Postulate)

Therefore Now, if you place the two circles on the top of each other, you will see that they coincide and hence .

**Hence Proved.**

*Note: In the same triangle if subtend equal angles at the center, then also the arcs are equal. For example:*

**Theorem 15: ***In equal circles, if the two arcs are equal, they would subtend equal angles at the center.*

Given: Two circles with centers respectively.

subtend at the respective centers.

To Prove:

Proof: (radius of equal circles)

Therefore, if you were to place the circles over each other, they will coincide. This also makes that .

**Hence Proved.**

**Theorem 16:*** In equal circles, if two chords are equal, they will cut equal arcs.*

Given: Two circles with centers respectively. Also

To Prove:

Proof: Consider

(radius of equal circles)

(radius of equal circles)

(Given)

Therefore (S.S.S postulate)

Therefore ( In equal circles, if two arcs subtends equal angles at the center, then the arcs are equal.)

**Hence Proved.**

**Theorem 17:** *In two equal circles, if the two arcs are equal the chords of the arcs are also equal.* (Converse of Theorem 16)

Given: Two circles with centers respectively. Also

To Prove:

Proof: Since

(In equal circles, if the two arcs are equal, they would subtend equal angles at the center)

Consider

(radius of equal circles)

(radius of equal circles)

**Hence Proved.**