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. 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

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A
Given
B
Definition of Center of a Circle
C
Definition of a point

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

Reason 2

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A
Definition of a line segment.
B
Two points determine a line.
C
Line segment postulate.

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

Reason 3

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

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

Reason 4

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A
Line segment are congruent.
B
Chords of a circle are congruent.
C
Radii of a circle are congruent.

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

Reason 5

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A
Radii of a circle are congruent.
B
Line segment are congruent.
C
Chords of a circle are congruent.

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

Reason 6

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A
Line segment are congruent.
B
Given
C
Radii of a circle are congruent

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Statement 7:△ FOG ≅ △ BOY

Reason 7

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A
ASA Congruence Postulate.
B
SAS Congruence Postulate.
C
SSS Congruence postulate

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Statement 8: ∠ FOG ≅ ∠ BOY

Reason 8

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

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

Reason 9

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

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

Reason 10

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

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Statement 11. arc FG ≅ arc BY

Reason 11

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A
Transitive Property.
B
Substitution Property.
C
Reflexive Property.

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With total honesty, did you get a perfect score?

Click the appropriate square below for your answer.

MBI_THE PROOF FINDER: Chord Arc Theorem

Chord Arc Theorem

DIRECTIONS:

Statement 1: Point O is the center

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

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

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

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

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

Statement 7:△ FOG ≅ △ BOY

Statement 8: ∠ FOG ≅ ∠ BOY

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

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

Statement 11. arc FG ≅ arc BY

With total honesty, did you get a perfect score?

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