If a quadrilateral is a parallelogram, the diagonals form two congruent triangles - How can I prove this
[From: ] [author: ] [Date: 12-05-01] [Hit: ]
Therefore BC AD, and AB = CDAngle AMD = angel CMD (opposite angles)And triangle AMD is similar to triangle CMB (equal angless) and congruent since they also have equal corresponding sides AD and CBTherefore AM = CM and BM = DM (corresponding sides of congruent triangles).Heres the diagram I used:http://i276.photobucket.com/albums/kk2/f…-this is clear that opposite sides of a param. are congruent and use alternate interior angles.......
angle ABD = angle CDB (same reason)
angle AMB = angle CBD (opposite angles)
angle DBC = angle BDA (alternate interior angles formed by transveral across 2 parallel lines)
Therefore angle ABC = angle CDA (each is sum of two equal angles).
angle BCD = angle BAD (alternate interior angles formed by transveral across 2 parallel lines)
So triangle ABC is similar to triangle CDA since they have equal angles. Since they share a corresponding side, AC, they are congruent.
Therefore BC AD, and AB = CD
Angle AMD = angel CMD (opposite angles)
And triangle AMD is similar to triangle CMB (equal angless) and congruent since they also have equal corresponding sides AD and CB
Therefore AM = CM and BM = DM (corresponding sides of congruent triangles).
Here's the diagram I used: http://i276.photobucket.com/albums/kk2/f…
-
this is clear that opposite sides of a param. are congruent and use alternate interior angles.
-
Note that before you can use SSS, ASA, etc. to prove triangles congruent, you have to first establish that the appropriate three pairs of corresponding sides and/or angles are congruent.
In this proof, the idea is to first start off with the definition of a parallelogram and use angle relationships for parallel lines and a transversal. These angle relationships, along with making use of the reflexive property for parts that triangles have in common, leads to proving triangles congruent (using SSS, SAS, etc). This leads to proving other angle or side congruences (by a combination of CPCTC and angle relationships for parallel lines), followed by proving a second pair of triangles congruent (using SSS, SAS, etc), followed by proving other angle or side congruences (by CPCTC).
Note: It is useful to mark the diagram with angle and side congruences, as the proof progresses.
Given: ABCD is a parallelogram, with diagonals AC and BD intersecting at M
keywords: two,this,parallelogram,form,can,is,If,How,triangles,congruent,prove,quadrilateral,diagonals,the,If a quadrilateral is a parallelogram, the diagonals form two congruent triangles - How can I prove this
