Below we write the formula to find the arc length of arc JK and arc NG.
Arc Length of JK
Arc Length of NG
mJK/360^(∘)* 2π* FH
mNG/360^(∘)* 2π* FG
Because FG≅ GH, we have that FG = GH. Then, by the Segment Addition Postulate we can get the following equation.
FH = FG+ GH ⇒ FH = 2 FG
Also, recall that the arc measure is equal to the measure of the central angle.
mNG = m∠ NFG mJK = m ∠ JFK
Let's substitute the new information into each formula.
Arc Length of JK
Arc Length of NG
m ∠ JFK/360^(∘)* 2π* 2 FG
m∠ NFG/360^(∘)* 2π* FG
Notice that we are also given that ∠ JFK ≅ ∠ KFJ. Then, by the Angle Addition Postulate we can write the following expression for ∠ JFL.
m∠ JFL = m ∠ JFK + m ∠ KFL
⇓
m∠ JFL = 2m ∠ JFK
Additionally, we have that ∠ JFL and ∠ NFG are vertical angles and the Vertical Angles Congruence Theorem tells us that they are congruent. Since congruent angles have the same measure, we can write the following equation.
m∠ NFG = m∠ JFL
⇓
m∠ NFG = 2m ∠ JFK
Finally, let's substitute the expression above into the expression on the right-hand column of the table above.
Arc Length of JK
Arc Length of NG
m ∠ JFK/360^(∘)* 2π* 2 FG
2m ∠ JFK/360^(∘)* 2π* FG
Notice that both expressions above are exactly the same, which shows that JK and NG have the same length.
Two-Column Proof
Given: & FG≅ GH, ∠ JFK ≅ ∠ KFL
Prove: & arc length of NG = arc length of JK
We will summarize the proof we just did in the following two-column table.
Statements
Reasons
1.
FG≅ GH, ∠ JFK ≅ ∠ KFL
1.
Given
2.
FG = GH
2.
Definition of congruent segments
3.
FH = FG+GH
3.
Segment Addition Postulate
4.
FH = 2FG
4.
Substitution
5.
m∠ JFK = m∠ KFL
5.
Definition of congruent angles
6.
m∠ JFL = m∠ JFK + m∠ KFL
6.
Angle Addition Postulate
7.
m∠ JFL = 2m∠ JFK
7.
Substitution
8.
∠ NFG ≅ ∠ JFL
8.
Vertical Angles Congruence Theorem
9.
m∠ NFG = m∠ JFL
9.
Definition of congruent angles
10.
m∠ NFG = 2m∠ JFK
10.
Substitution
11.
arc length of JK = m∠ JFK/360^(∘)* 2π FH arc length of NG = m∠ NFG/360^(∘)* 2π FG
11.
Formula for arc length
12.
arc length of JK = m∠ JFK/360^(∘)* 2π (2FG) arc length of NG = 2m∠ JFK/360^(∘)* 2π FG
