Contradictions and Proofs in Geometry

This video tutorial focuses on indirect proofs in geometry, using two-column proofs to demonstrate the process. It begins with an example involving two triangles and guides viewers through the steps of making assumptions, identifying contradictions, and reaching a conclusion. A second example is provided for practice, reinforcing the method of indirect proof by assuming the negation of the statement to be proven and finding contradictions.

What is the primary goal of an indirect proof?

• To directly prove the given statement

• To assume the negation of the statement and find a contradiction

• To prove the statement by using examples

• To use only direct evidence to support the statement

In the given problem, what is the initial assumption made for the indirect proof?

• Angle B is not congruent to angle E

• Angle B is congruent to angle E

• Triangle ABC is congruent to triangle DEF

• AC is congruent to DF

Which postulate is used to prove the congruence of triangles ABC and DEF?

• Angle-Side-Angle (ASA)

• Side-Side-Side (SSS)

• Side-Angle-Side (SAS)

• Angle-Angle-Side (AAS)

What is the contradiction found in the proof of the first problem?

• AC is congruent to DF

• Angle B is congruent to angle E

• AC is not congruent to DF

• Triangles ABC and DEF are not congruent

What is the conclusion drawn from the contradiction in the first problem?

• Angle B is not congruent to angle E

• Angle B is congruent to angle E

• Triangles ABC and DEF are congruent

• AC is congruent to DF

In the new problem, which two sides are given as congruent?

• AC and DC

• BD and EC

• BC and EC

• AB and ED

What is the task in the new problem involving triangles ABC and EDC?

• Prove that AC is congruent to EC

• Prove that angle B is not congruent to angle D

• Prove that angle B is congruent to angle D

• Prove that triangles ABC and EDC are congruent

Which postulate is used to prove the congruence of triangles ABC and EDC in the new problem?

• Angle-Angle-Side (AAS)

• Side-Angle-Side (SAS)

• Side-Side-Side (SSS)

• Angle-Side-Angle (ASA)

What is the contradiction found in the new problem?

• AC is not congruent to EC

• Triangles ABC and EDC are not congruent

• Angle B is congruent to angle D

• AC is congruent to EC

What is the conclusion drawn from the contradiction in the new problem?

• AC is congruent to EC

• Angle B is not congruent to angle D

• Angle B is congruent to angle D

• Triangles ABC and EDC are congruent