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  1. In the given circle, segments OA, OB, OC and OD are radius of the circle.

    In other words, all those segments are congruent.

    From that we can deduct that triangles OAB and OCD are isosceles triangles, because all segments are equal. That makes those triangles congruent, because they have the same sides.

    From the congruence we can deduct that all corresponding internal angles inside those triangles are actually congruent, that's why angle OAB and ODC must be congruent.

    Also, angles DOC and BOA are also congruent, which means their subtended arcs are also congruent, because they have the same radius, so mAB = mCD.

  2. The approximated length of arc x is 4 inches ⇒ C

    Step-by-step explanation:

    The diameter of the outer circle is 28 inchesThere is a boundary with width 2 inches

    That means the diameter of the inner circle is less than the diameter of the outer circle by 4 inches (2 in from each side)

    The diameter of the inner circle = 28 - 4 = 24 inches

    The circumference of the inner circles formed from 20 congruent arcs of length x inches

    The circumference of the inner circle = 20 x

    Now let us find x

    The formula of the circumference of a circle is C = π d, where d is the diameter of the circle

    ∵ The diameter of the inner circle is 24 inches

    ∴ d = 24

    ∴ C = 24 π

    ∵ C = 20 x

    - Equate 20 x by 24 π

    ∴ 20 x = 24 π

    - Divide both sides by 20

    ∴ x = 3.769911184

    - Round it to the nearest unit

    ∴ x ≅ 4

    The approximated length of arc x is 4 inches

  3. OF

    Step-by-step explanation:

    Since the arcs are congruent, the lines intercepting them are congruent, and you will get 2 congruent right triangles. So OA is congruent to OF by CPCTC.

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