What is the reason for statement 5 in this proof?

Given: ΔABC, where AB = BC (view diagram)
Prove: m∠BAC=m∠BCA

Statement
Reason
1. Let ΔABC be an isosceles triangle with
AB = BC.

given

2. Create point D on side AC¯¯¯¯¯ so BD¯¯¯¯¯ bisects
∠ABC.

constructing an angle bisector

3. m∠ABD=m∠CBD

definition of angle bisector

4. BD = BD

Reflexive Property of Equality

5. ΔABD≅ΔCBD

6. m∠BAC=m∠BCA

Corresponding angles of congruent triangles have equal measures.

A. ASA

B. SSS

C. AAS

D. SAS

You are comparing side AB, angle ABD, and side BD in one triangle to side CB, angle CBD, and side BD in the other triangle. That is, you are comparing a Side, Angle, and Side in each triangle. The SAS postulate is the reason the triangles are congruent.

Answer Link

FelisFelis FelisFelis · Answer 2 · 2019-11-27T05:18:41+01:00

Answer:

The correct option is D) SAS

Step-by-step explanation:

Consider the provided statement.

SAS Similarity Theorem: If two sides of a triangle are proportional to the two sides of another triangle and the included angle in both are congruent, then the two triangles are similar.

Statement 1: Let ΔABC be an isosceles triangle where AB = BC.

Reason 1: Given

Statement 2: Create point D on so that bisects ∠ABC as shown.

Reason 2: Constructing an angle bisector.

Statement 3: m∠ABD = m∠DBC

Reason 3: Definition of angle bisector

Statement 4: BD = BD

Reason 4: Reflexive Property of Equality

Statement 5: ΔABD ≅ ΔCBD

Reason 5: SAS

Statement 6: m∠BAC = m∠BCA

Reason 6: Corresponding angles of congruent triangles are equal.

From Reflexive Property of Equality we know BD=BD, m∠ABD = m∠DBC definition of angle bisector and AB = BC, which follows the Side angle side (SAS) similarity.