MBI_THE PROOF FINDER: Chord Arc Theorem

Chord Arc Theorem

If two chords in a circle are congruent, then their intercepted arcs are congruent.

DIRECTIONS:

This proving activity is a sort of game. If you got a perfect score, you will earn the MBI Agent Badge. MBI stands for Math Bureau of Investigation. It is related to investigation because every statement must be provided with accurate reason in order to prove the theorem correctly.[br][br]Given the circle below, point O is the center and line segment FG and line segment BY are chords, and they are congruent (≅).Prove that arc FG ≅ arc BY.

Statement 1: Point O is the center

Reason

Definition of a point Given Definition of Center of a Circle

Statement 2: Draw line segment OF, line segment OG, line segment OB and line segment OY.

Reason 2

Line segment postulate. Two points determine a line. Definition of a line segment.

Statement 3: Line segment OF, line segment OG, line segment OB and line segment OY are radii.

Reason 3

Definition of radius/radii of a circle A radius is a line segment. Radius/radii of a circle postulate.

Statement 4: Line segment OF is congruent to line segment OG.

Reason 4

Radii of a circle are congruent. Line segment are congruent. Chords of a circle are congruent.

Statement 5: Line segment OG is congruent to line segment OY.

Reason 5

Radii of a circle are congruent. Chords of a circle are congruent. Line segment are congruent.

Statement 6: Line segment FG is congruent to line segment BY.

Reason 6

Line segment are congruent. Given Radii of a circle are congruent

Statement 7:△ FOG ≅ △ BOY

Reason 7

SSS Congruence postulate ASA Congruence Postulate. SAS Congruence Postulate.

Statement 8: ∠ FOG ≅ ∠ BOY

Reason 8

Corresponding Parts of Congruent Triangles are Congruent ( CPCTC). Corresponding Parts of Acute Triangles are Congruent ( CPATC). Vertical Angles are Congruent ( VAC).

Statement 9: ∠ FOG and ∠ BOY are central ∠s.

Reason 9

Definition of central angles. Vertical angles in a circle form central angles. Central angles form vertical angles.

Statement 10: m∠ FOG = measure arc FG ,and m∠ BOY = measure arc BY.

Reason 10

The measure of the central angle is equal to the measure of the intercepted arc. The measure of the central angle is equal to the measure of the major arc. The measure of central angle is equal to the measure of the minor arc.

Statement 11. arc FG ≅ arc BY

Reason 11

Reflexive Property. Transitive Property. Substitution Property.

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