We will prove the Inscribed Angle Theorem, Corollary 3.
2
&Given:&& Quadrilateral ABCD
& &&inscribed in ⊙ O
&Prove:&& ∠ A and ∠ C are supplementary.
& && ∠ B and ∠ D are supplementary.
Since we want to show that two pairs of angles are supplementary, we will investigate each pair separately.
Let's start with ∠ A and ∠ C!
∠ A and ∠ C
We will begin by considering the angles and their arcs.
We know that, by the Inscribed Angle Theorem, m∠ A is half mBCD and m∠ C is half mBAD.
m∠ A=1/2 mBCD
m∠ C=1/2 mBAD
We can combine these two equalities by the Addition Property of Equality.
m∠ A+ m∠ C=1/2 mBCD+1/2 mBAD
Now, we can factor out 12 and have the sum of the two arcs.
m∠ A+ m∠ C=1/2( mBCD+ mBAD)
Notice that the sum of the arcs gives us a whole circle. Since the arc measure of a circle is 360 ^(∘), the sum of mBCD and mBAD is 360 ^(∘).
mBCD+ mBAD=360^(∘)
Next, we will multiply the above equality by 12 by the Multiplication Property of Equality.
1/2( mBCD+ mBAD)=180^(∘)
With this, we know that the sum of m∠ A and m∠ C is 180^(∘) by the Substitution Property.
m∠ A+ m∠ C=180^(∘)
Finally, by the definition of supplementary angles, we can complete the first part of the proof.
∠ A and ∠ C are supplementary.
Let's summarize the above process in a flow proof.
∠ B and ∠ D
We can prove that ∠ B and ∠ D are supplementary by proceeding in the same way.
Again, we can show the process in a flow proof.
Conclusion
Combining the two parts, we can conclude that ∠ A and ∠ C are supplementary, as well as ∠ B angle ∠ D, given that ABCD is a quadrilateral inscribed in ⊙ O.
