Jeremy's proof demonstrates that congruent arcs lead to congruent central angles, and because all radii in a circle are congruent, we can conclude by SAS that the triangles formed are congruent and thus the intercepted chords are congruent by CPCTC.
We can summarize Jeremy's proof that congruent arcs are intercepted by congruent chords in a circle by arranging the given statements correctly.
Based on the summary provided, Jeremy's statements can be ordered as follows:
Given that two arcs of a circle are congruent, their measures are equal by the definition of congruence.
C. angle AOC ≅ angle BOD - Central angles are equal to their intercepted arcs, and so the two central angles are congruent by the definition.
6. AO ≅ BO and CO ≅ DO - Since all radii of a circle are congruent (AO to BO and CO to DO are equal).
A. Triangle AOC ≅ Triangle BOD - The two triangles are congruent by the Side-Angle-Side (SAS) postulate.
8. AC ≅ BD - Finally, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), the chords AC and BD are congruent.
This step-by-step proof shows how the congruence of chords is directly related to the congruence of arcs and central angles in a circle.
The probable question may be:
Jeremy used the figure and the given summary to prove that congruent arcs are intercepted by congruent chords.
In circle O, Radius OD and OB are drawn the point D and B are join which form a triangle ODB.
Radius OA and OC are drawn the point A and C are join which form a triangle OAC.
Given that two arcs of a circle are congruent, their measures are equal by the definition of congruence. Central angle measures are equal to their intercepted arcs, so by the transitive property, the two central angle measures are equal. By definition, the two angles are also congruent. Since all radii are congruent, the two triangles are congruent by SAS. Finally, the intercepted chords are congruent by CPCTC.
Drag the statements to the positions that match the summary of Jeremy's proof.
Statements
1.________
2. _________
3. MAC=m angle AOC and mBD = m angle BOD
4. ___________
5. _________
6. AO ≅ BO and CO ≅ DO
7.___________
8. AC ≅ BD
A. Triangle AOC ≅ Triangle BOD
B. mAC = mBD
C. angle AOC ≅ angle BOD
D. AC ≅ BD
E. m angle AOC=m angle BOD
