Question

In isosceles triangle ∆ABC, BM is the median to the base AC . Point D is on BM . Prove the following triangle congruencies:∆ABD ≅ ∆CBD

Answer 1

By utilizing the properties of isosceles triangles and medians, we can prove that ▲ABD and ▲CBD are congruent through the Side-Angle-Side (SAS) congruence criterion.

Step-by-step explanation:

To prove that triangles ▲ABD and ▲CBD are congruent in an isosceles triangle ▲ABC with BM as the median to the base AC, we begin by noting that as BM is the median, it splits AC into two equal parts, meaning AM = MC. Since ▲ABC is isosceles, AB = BC by definition. Now, consider the two triangles ▲ABD and ▲CBD.

• Triangle ▲ABD has sides AB and AD, and triangle ▲CBD has sides CB and CD.
• Since AB = BC and AD = CD (because BM is a median, it splits BM into BD and DM), we have two sides of each triangle being equal.
• Because BM is a median, ∠ABM is equal to ∠CBM (base angles in an isosceles triangle are equal).
• Therefore, by Side-Angle-Side (SAS) congruence criterion, ▲ABD ≅ ▲CBD.

Answer 2

We know that triangle ABD and triangle CBD are congruent because of SAS.

Step-by-step explanation:

AB is congruent to BC because of the definition of an isosceles triangle

BD=BD because of the reflexive property

m<ABD=m<CBD because BM is the median of an isosceles triangle

Thus, triangle ABD is congruent to triangle CBD because of SAS