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 }
You can't use SAS, since you know nothing about any of the angles
BUT, you can use SSS to prove
∆ ABD congruent to ∆ CBD
Therefore corresponding angles are equal, that is
∠ABD = ∠CBD and ∠ADB = ∠CDB by SSS
oops sorry
angle abd=angle1
angle adb=angle 3
angle cbd=angle 2
angle bdc=angle 4
In ∆ABD, we know that AB=AD, so ∡D=∡B
Since BD bisects ∡ADC, in BDC, ∡B=∡D, so DC=BC. Since the two triangles share side BD, they are congruent by ASA or SAS
To prove that BD bisects both angles ABC and ADC using the SAS congruence approach, we need to show that triangles ABD and CBD are congruent, as well as triangles BCD and ABD.
Given:
1. AB = BC (given)
2. AD = CD (given)
To prove:
1. ∠ABD ≅ ∠CBD (BD bisects ∠ABC)
2. ∠CBD ≅ ∠ADB (BD bisects ∠ADC)
Proof:
To prove that triangle ABD ≅ triangle CBD, we will use the following congruence criteria:
1. Side-Side-Side (SSS) congruence:
If three sides of a triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
2. Side-Angle-Side (SAS) congruence:
If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.
Now, let's break down the proof:
1. AB = BC (given)
2. AD = CD (given)
3. BD (common side)
4. ∠ABD and ∠CBD (common angles)
By SAS congruence, we know that triangles ABD and CBD are congruent. Therefore, we can conclude that ∠ABD ≅ ∠CBD.
Similarly,
1. AB = BC (given)
2. AD = CD (given)
3. BD (common side)
4. ∠ADC and ∠ADB (common angles)
By SAS congruence, we can again conclude that triangles BCD and ABD are congruent. Therefore, ∠CBD ≅ ∠ADB.
Hence, we have proved that BD bisects both angles ABC and ADC using the SAS congruence approach.