If is the midpoint of [segment] BD¯ ,AC= 2 feet , and AE= 2 , which answer choices can be used to show △ABC≅△ADE by the SAS Congruence Postulate?

Question

A window design calls for two panes of glass, as shown.

If  is the midpoint of [segment] BD¯ ,AC= 2 feet , and AE= 2 , which answer choices can be used to show △ABC≅△ADE  by the SAS Congruence Postulate?

There is more than one correct answer choice. Select all correct answer choices

​1) DE=3 feet and BC=3 feet

2) Definition of congruent triangles

3)Definition of congruent segments

4)Reflexive Property of Congruence

5)Definition of congruence

6)Definition of midpoint

7)Vertical Angle Theorem

I'm not entirely sure how I am supposed to figure this out.

Answer 1

Hi Klauss,

First we can use the pieces of information they gave us.

Since A is the “midpoint” of BD, we know that the length of AB equals the length of AD. This is because the definition of midpoint says that a midpoint is the same distance from both endpoints. This also means that we have figured out the first “S” in the SAS congruence proof, because when two segments are the same length they meet the definition of congruent segments.

Since we were also told that the lengths of AC and AE were both equal to 2, we can again say these are congruent because of the definition of congruent segments. This is a second “S” from the proof.

One thing that matters in an SAS proof is that whatever angle we are trying to prove congruent, must be between the two “S” sides. In the left triangle that angle would be angle DAE, and in the right triangle would be angle BAC. These are what we would call vertical angles (angles formed by a pair of intersecting lines, in this case lines DB and EC). The vertical angle theorem says that vertical angles are congruent, so we have proved our “A.”

Therefore triangles ABC and ADE are congruent by the SAS Congruence Postulate.

Please let me know if I can be more helpful, thanks!

Joe