It is given that the circles on the diagram are congruent.

∠ T F E

and

∠ M K L

are congruent. Also,

∠ T F E

and

H F G,

as well as

∠ M K L

and

∠ J K N,

are vertical angles. By the Vertical Angles Theorem these four angles are congruent.

∠ H F G ≅ ∠ T F E ≅ M L K ≅ J K N

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 $\overline{T E},$ $\overline{H G},$ $\overline{M L},$ and $\overline{J N}$ 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

\overline{T E},

\overline{H G},

\overline{M L},

and

\overline{J N}

are congruent, their arcs

T E,

H G,

M L,

and

J N

are also congruent.

T E ≅ H G ≅ M L ≅ J N

Similarly, angles

∠ T F H

and

∠ E F G,

just as

∠ M K J

and

∠ N K L,

are also vertical angles and therefore congruent.

Note that

∠ T F E

and

∠ T F H

and

∠ M K L

and

∠ M K J

are two pairs of adjacent supplementary angles. In other words, the sum of their measures is

18 0^{\circ} .

Since

m ∠ T F E = m ∠ M K L,

we can conclude the following.

m ∠ T F E + m ∠ T F H = 18 0^{\circ} m ∠ M K L + m ∠ J K M = 18 0^{\circ} ⇓ m ∠ T F H = m ∠ J K M

All four angles —

∠ T F E,

∠ T F H,

∠ M K L,

and

∠ M K J

— are congruent angles.

Now, let's recall Theorem $12$ - $4 .$

Within a circle or in congruent circles, congruent central angles have congruent arcs.

By the theorem, arcs

T F,

E G,

M J,

and

N L

are congruent arcs.

T F ≅ E G ≅ M J ≅ N L

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