Given: segment AB is congruent to segment BC; angle 1 is congruent to angle 2
Prove: triangle ABC is congruent to triangle DEC
There is no info on DEC. Do you have a diagram that you refer to?
To prove that triangle ABC is congruent to triangle DEC, we can use the Side-Angle-Side (SAS) congruence theorem.
1. Given: segment AB is congruent to segment BC and angle 1 is congruent to angle 2.
2. Draw segment AD connecting the vertices of triangle ABC to point D on side AC.
3. Now, we have to prove that segment DE is congruent to segment DC and angle D is congruent to angle C.
4. From step 2, we have triangle ABD and triangle CBD.
5. By the given information, we know that segment AB is congruent to segment BC, so we have side AB congruent to side BC.
6. Given that angle 1 is congruent to angle 2, we have angle ABD congruent to angle CBD.
7. Now, for triangles ABD and CBD, we have side AB congruent to side BC and angle ABD congruent to angle CBD, so we have side-angle-side congruence.
8. Using the SAS congruence theorem, we can conclude that triangle ABD is congruent to triangle CBD.
9. From triangle ABD, we have segment AD congruent to segment CD (by CPCTC - corresponding parts of congruent triangles are congruent).
10. Therefore, we can conclude that segment DE is congruent to segment DC, as AD is congruent to CD.
11. Additionally, angle D is congruent to angle C, as they are corresponding angles of congruent triangles ABD and CBD.
12. By the SAS congruence theorem, triangle ABC is congruent to triangle DEC, as we have side AB congruent to side DE, angle A congruent to angle D, and side BC congruent to side DC.
13. Therefore, we have proven that triangle ABC is congruent to triangle DEC.
To prove that triangles ABC and DEC are congruent, we can use the Side-Angle-Side (SAS) congruence criterion. This criterion states that if two sides of one triangle are congruent to two corresponding sides of another triangle, and the included angles are congruent, then the triangles are congruent.
In this case, we are given that segment AB is congruent to segment BC, and angle 1 is congruent to angle 2. To show congruence, we need to find a third corresponding side and angle that are congruent between the two triangles.
To find the third corresponding side, notice that segment AC can be used. Since segment AB is congruent to segment BC, and segment AB and segment AC are two corresponding sides of triangle ABC, segment AC must also be congruent to segment BC.
Now, we need to find the third corresponding angle. Since we know that angle 1 is congruent to angle 2, and angle 1 and angle 3 are adjacent angles in a straight line, we conclude that angle 3 is congruent to angle 2 as well.
Therefore, we have the three congruent corresponding parts: segment AB is congruent to segment BC (side), angle 1 is congruent to angle 2 (angle), and segment AC is congruent to segment BC (side). By the SAS congruence criterion, we can conclude that triangle ABC is congruent to triangle DEC.