In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.
Considering this theorem, our goal will be showing that ∠ APB ≅ ∠ CPD. Let's start! Notice that AP, BP, CP, and DP are radii of ⊙ P by the definition of a radius.
AP, BP, CP, and DP
are radii of ⊙ P.
Note that all radii of a circle are congruent.
AP ≅ BP ≅ CP ≅ DP
With this information and given that AB≅ CD, we have three sides of △ APB that are congruent to the corresponding three sides of △ CPD.
Therefore, by the Side-Side-Side (SSS) Congruence Theorem △ APB is congruent to △ CPD.
△ APB ≅ △ CPD
We know that the corresponding parts of congruent triangles are congruent (which can be abbreviated as CPCTC).
From here, we can conclude that ∠ APB and ∠ CPD are congruent.
∠ APB ≅ ∠ CPD
Finally, by the Congruent Central Angles Theorem (Theorem 10.4) we can say that AB and CD are congruent.
AB≅ CD
Let's summarize the above process in a flow proof.
Given:& AB and CD are congruent chords.
Prove:& AB ≅ CD
Proof:
b In this part, we will prove that AB≅ CD given that AB ≅ CD.
We will write a proof similar to the one in Part A.
Given:& AB ≅ CD
Prove:& AB ≅ CD
This time, we will proceed by showing that △ APB and △ CPD are congruent by the Side-Angle-Side (SAS) Congruence Theorem. Let's do it!
Here, we first proved that △ APB ≅ △ CPD by the SAS Congruence Theorem.
Then, since the corresponding parts of congruent triangles are congruent, we finally showed that AB≅ CD.
