given: segment AB is paralell to segment DC; segment AB is congruent to segment to DC
prove: triangle ABC is congruent to triangle CDA
statements:
1. segment AB is congruent to segment DC
2.segment AC is congruent to segment AC
3.segment AB is paralell to segment DC
4.angle BAC is congruent to angle DCA
5.triangle ABC is congruent to triangle CDA
(i need to find the reasons)???
reasons:
1.?
2.?
3.?
4.?
5.?
It is hard to provide geometry assistance without a figure to work with. Your problem seems to be representable by a parallelogram ABCD, where AC is a diagonal. The angle BAC is the same as the angle ACD, because they are formed by a line intersecting parallel lines AB and CD. (There is a theroem about that). That means triangles ABC and ACD have two pairs of sides (AB/CD and the common side AC)with an included angle congruent. That makes the triangles congruent. (Side-Angle-Side theorem).
What is the theorem used
Reasons:
1. Given: segment AB is congruent to segment DC (Given information).
2. Reflexive property of congruence: segment AC is congruent to segment AC (A segment is congruent to itself).
3. Given: segment AB is parallel to segment DC (Given information).
4. Corresponding angles postulate: angle BAC is congruent to angle DCA (If two lines are parallel, then the corresponding angles formed by a transversal are congruent).
5. Side-Angle-Side (SAS) congruence theorem: triangle ABC is congruent to triangle CDA (If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent).
Note: The congruence between segments AB and DC is not sufficient to prove the congruence of triangles ABC and CDA. The additional information of the parallel lines and congruent angles is required.
Reasons:
1. Segment AB is congruent to segment DC (given).
2. Segment AC is congruent to segment AC (reflexive property).
3. Segment AB is parallel to segment DC (given).
4. Angle BAC is congruent to angle DCA (corresponding angles formed by parallel lines).
5. By SAS (Side-Angle-Side), triangle ABC is congruent to triangle CDA.
Since two sides and the included angle of triangle ABC are congruent to the corresponding parts of triangle CDA, the two triangles are congruent.
Therefore, triangle ABC is congruent to triangle CDA.