Give a two-column proof for the following:9. Given: 2𝑥 − 5(𝑥 + 3) = 9 + 𝑥Prove: 𝑥 = −610.Given: 𝑚∠𝐸𝐵𝐶 = 𝑚∠𝐸𝐶𝐵Prove: ∠𝐸𝐵𝐴 ≅ ∠𝐸𝐶𝐷

Sure, here are the two-column proofs for the given problems:

Problem 1: Given: 2x - 5(x + 3) = 9 + x Prove: x = -6

Steps:

1. Distribute -5 in the expression -5(x + 3) to get -5x - 15. Statement: 2x - 5x - 15 = 9 + x Reason: Distributive Property

2. Combine like terms on the left side to get -3x - 15. Statement: -3x - 15 = 9 + x Reason: Combining Like Terms

3. Subtract x from both sides to get -4x - 15 = 9. Statement: -4x - 15 = 9 Reason: Subtraction Property of Equality

4. Add 15 to both sides to get -4x = 24. Statement: -4x = 24 Reason: Addition Property of Equality

5. Divide both sides by -4 to get x = -6. Statement: x = -6 Reason: Division Property of Equality

Problem 2: Given: m∠EBC = m∠ECB Prove: ∠EBA ≅ ∠ECD

Steps:

1. If m∠EBC = m∠ECB, then ∠EBC ≅ ∠ECB. Statement: ∠EBC ≅ ∠ECB Reason: Definition of Congruent Angles

2. If ∠EBC ≅ ∠ECB, then their supplements are congruent. Statement: ∠EBA ≅ ∠ECD Reason: Congruent Supplements Theorem

Please note that the second proof assumes that ∠EBA and ∠EBC are supplementary, as well as ∠ECD and ∠ECB. If this is not the case, the proof would not hold.