ABCD is a quadrilateral in which AB=BC and AD =CD ,Show that BD bisects both the angles ABC and ADC {by using SAS congruence }
Given in figure
angle abd=angle1
angle adb=angle 3
angle cbd=angle 2
angle bdc=angle 4
This changes nothing in my solution I gave you earlier.
You are simply giving the angles nice short names.
To show that BD bisects both angles ABC and ADC using SAS congruence, we need to prove that triangle ABD is congruent to triangle CBD and triangle ABD is congruent to triangle CBD. This will imply that angle ABC is equal to angle BCD, and angle ADC is equal to angle ABD.
Here's how we can prove it:
Step 1: Given the information that AB=BC and AD=CD, we can conclude that sides AB and AD are congruent to sides BC and CD, respectively.
Step 2: We are also given that angle ABD (angle 1) is equal to angle CBD (angle 2).
Step 3: By SAS (Side-Angle-Side) congruence, we need to prove that the following conditions are satisfied for the triangles ABD and CBD:
i) Side AB is congruent to side CB (already given).
ii) Angle ABD is congruent to angle CBD (already given).
iii) Side AD is congruent to side CD (already given).
Step 4: Since all the three conditions for congruence (mentioned in Step 3) are satisfied, triangle ABD is congruent to triangle CBD by the SAS congruence criterion.
Step 5: As a consequence of Step 4, we can conclude that angle ABC is equal to angle BCD. This is because corresponding parts of congruent triangles are equal.
Step 6: Similarly, we can prove that triangle ABD is congruent to triangle CBD by SAS congruence criterion.
Step 7: As a consequence of Step 6, we can conclude that angle ADC is equal to angle ABD.
Therefore, we have shown that BD bisects both angles ABC and ADC using SAS congruence.