Investigate each statement of the given paragraph proof separately. The second statement is little bit tricky. Isolate the angle measures to catch the theorem.
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Considering the Given and Prove statements, we can say that we will prove the conditional statement of the theorem.
Given:& ⊙ C with inscribed
&quadrilateral DEFG
Prove:& m∠ D + m∠ F = 180 ^(∘),
&m∠ E + m∠ G = 180^(∘)
To complete the paragraph proof, we will use following diagram.
We will complete the proof by investigating each statement of the proof separately.
Statement I
The first statement uses the Arc Addition Postulate (Postulate 10.1).
By the Arc Addition Postulate,
mEFG+ =360^(∘)
and mFGD+mDEF=360^(∘).
With the postulate we are looking for an arc that completes EFG to a full circle.
As we can see, EFG and EDG form a full circle. Therefore, we can complete the first statement as follows.
By the Arc Addition Postulate,
mEFG+mEDG=360^(∘)
and mFGD+mDEF=360^(∘).
Statement II
For this statement we will determine the theorem which gives us mEDG=2m∠ F, mEFG=2m∠ D, mDEF=2m∠ G, and mFGD=2m∠ E. To do so, let's begin by isolating the angle measures.
m∠ F=1/2mEDG& m∠ D=1/2mEFG
m∠ G=1/2mDEF& m∠ E=1/2mFGD
From here we can conclude that the measure of each inscribed angle is half the measure of its intercepted arc.
Therefore, the theorem that completes the statement is the Inscribed Angle Theorem.
Using the Inscribed Angle Theorem,
mEDG=2m∠ F, mEFG=2m∠ D,
mDEF=2m∠ G, and mFGD=2m∠ E.
Statement III
For the third step of the proof, we will substitute 2m∠ D and 2m∠ F for mEFG and mEDG into mEFG+mEDG=360^(∘) using the Substitution Property of Equality. Then, by dividing each side of the equation by 2, we can complete the statement.
By the Substitution Property of Equality,
2m∠ D+2m∠ F=360 ^(∘),
so m∠ D+m∠ F=180 ^(∘).
Statement IV
In Statement III, we proved the first part of the Prove statement.
Prove:& m∠ D + m∠ F = 180 ^(∘),
&m∠ E + m∠ G = 180^(∘)
Therefore, proceeding in the same way, we can also prove the second part and complete Statement IV.
Similarly, m∠ E + m∠ G = 180^(∘).
Completed Proof
Finally, we can write the completed paragraph proof.
By the Arc Addition Postulate, mEFG+mEDG=360^(∘) and mFGD+mDEF=360^(∘). Using the Inscribed Angle Theorem, mEDG=2m∠ F, mEFG=2m∠ D, mDEF=2m∠ G, and mFGD=2m∠ E. By the Substitution Property of Equality, 2m∠ D+2m∠ F=360 ^(∘), so m∠ D+m∠ F=180 ^(∘). Similarly, m∠ E + m∠ G = 180^(∘).
