Fill In The Blanks
What are the missing reasons in the two column proof.
Given: AC = CD, and m<ACB is greater than the m<BCD
Prove: AB > BD
1. AC = CD 1. Response area
2. CB = CB 2. Response area
3. m<ACB is greater than the m<BCD 3. Response area
4. AB > BD 4. Response area
In a two-column proof, we need to list both the statements and the reasons for those statements. The reasons can include given information, definitions, postulates, or previously established theorems that are relevant to the argument. Given the statements in your question, here's how the proof might be filled out:
1. AC = CD 1. Given
2. CB = CB 2. Reflexive Property of Equality
3. m<ACB > m<BCD 3. Given
4. Side Opposite to the larger angle is greater 4. Theorem/Postulate/Definition
Explanation for Step 4 Reason:
If one angle of a triangle is larger than another, then the side opposite the larger angle is longer than the side opposite the smaller angle. This is sometimes referred to as the "Hinge Theorem" or the "Converse of the Isosceles Triangle Theorem."
So for step 4, the more exact phrasing of the reason might be:
4. If in a triangle, one side is greater than another, then the angle opposite the greater side is greater than the angle opposite the lesser side. (Hinge Theorem/Converse of the Isosceles Triangle Theorem)
Proving that "AB > BD" relies on the triangle inequality principle and the stated geometric properties. Since <ACB is greater than <BCD and both angles are in triangle ABC, by the Hinge Theorem, the side opposite the larger angle (<ACB), which is AB, must be longer than the side opposite the smaller angle (<BCD), which is BD.