It is given that the circles on the diagram are congruent.
As we can see, central angles ∠ TFE and ∠ MKL are congruent. Also, ∠ TFE and HFG, as well as ∠ MKL and ∠ JKN, are vertical angles. By the Vertical Angles Theorem these four angles are congruent.
∠ HFG≅ ∠ TFE≅ MLK ≅ JKN
Let's now review what the Theorem 12-5 states.
Theorem 12-5
Within a circle or in congruent circles, congruent central angles have congruent arcs.
According to this theorem, chords TE, HG, ML, and JN are congruent. Next we can use Theorem 12-6.
Theorem 12-6
Within a circle or in congruent circles, congruent chords have congruent arcs.
Since chords TE, HG, ML, and JN are congruent, their arcs TE, HG, ML, and JN are also congruent.
TE≅ HG≅ ML≅ JN
Similarly, angles ∠ TFH and ∠ EFG, just as ∠ MKJ and ∠ NKL, are also vertical angles and therefore congruent.
Note that ∠ TFE and ∠ TFH and ∠ MKL and ∠ MKJ are two pairs of adjacent supplementary angles. In other words, the sum of their measures is 180^(∘). Since m∠ TFE=m∠ MKL, we can conclude the following.
l m∠ TFE+m∠ TFH=180^(∘) m∠ MKL+m∠ JKM=180^(∘)
⇓
m∠ TFH=m∠ JKM
All four angles — ∠ TFE, ∠ TFH, ∠ MKL, and ∠ MKJ — are congruent angles.
