Question

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

Answer 1

The correct option is D) SAS

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.

Answer 2

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.