As we can see, ∠ BAD and ∠ ADC are alternate interior angles.
∠ BAD and ∠ ADC
are alternate interior angles.
By the Alternate Interior Angles Theorem we know that the alternate interior angles are congruent.
∠ BAD ≅ ∠ ADC
By the definition of congruent angles, ∠ BAD and ∠ ADC have the same angle measure.
m∠ BAD ≅ m∠ ADC
In the first part of our proof we showed that m∠ BAD ≅ m∠ ADC. Next, we will consider the Inscribed Angle Theorem.
By this theorem, we can conclude that m∠ BAD is half mBD and m∠ ADC is half AC.
m∠ BAD=1/2mBD
m∠ ADC=1/2 mAC
Now, since m∠ BAD=m∠ ADC, we can substitute m∠ BAD for m∠ ADC in m∠ ADC= 12mAC.
m∠ BAD=1/2mBD
m∠ BAD=1/2 mAC
Then, we can use the Transitive Property of Equality.
1/2 mAC =1/2mBD
By the Multiplication Property of Equality, we can multiply both sides of the above equation by 2 to eliminate the denominators.
mAC = mBD
Finally, remember that congruent arcs have the same arc measure.
Therefore, the arcs of a circle that are included between parallel chords are congruent.
AC ≅ BD
