Question

PLEASE! 14 points! Given B and G lie on a circle centered at G, and A and F lie on another circle also centered at C. prove triangle ABC is similar to triangle EDC. Use a geometric statement and proof - two column proof

Answer 1

Geometric statement: If B and G lie on a circle centered at C, and A and F lie on another circle also centered at C, then triangle ABC is similar to triangle EDC.

Proof:

Statement Reason

1. B and G lie on a circle centered at C. Given

2. A and F lie on another circle also centered at C. Given

3. ∠ABC = ∠EDC. Vertical angles are congruent.

4. ∠BGC = ∠AFC. Inscribed angles that intercept the same arc are congruent.

5. ∠ABC + ∠BGC = 180°. Angles around a point add up to 360°, and ∠BGC is supplementary to ∠ABC.

6. ∠EDC + ∠AFC = 180°. Angles around a point add up to 360°, and ∠AFC is supplementary to ∠EDC.

7. ∠BGC = ∠EDC. Substitution from step 4.

8. Triangle ABC is similar to triangle EDC. AA similarity criterion, using steps 3 and 7.

Therefore, triangle ABC is similar to triangle EDC, given that B and G lie on a circle centered at C, and A and F lie on another circle also centered at C.