Question

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unit 4: congruent triangles
homework 6: congruent triangle proofs:
asa, aas, and hl
this is a 2 - page document!
complete proofs 1 - 2 using angle - side - angle (asa).

1. given: \\( \overline{bd} \\) bisects \\( \angle abc, \angle bda \cong \angle bdc \\)

prove: \\( \triangle abd \cong \triangle cbd \\)

Explanation:

Step1: Definition of angle bisector

Since \(BD\) bisects \(\angle ABC\), we have \(\angle ABD\cong\angle CBD\) (by the definition of an angle bisector).

Step2: Given congruent angles

We are given that \(\angle BDA\cong\angle BDC\).

Step3: Common side

\(BD = BD\) (by the reflexive property of equality, which states that any segment is congruent to itself).

Step4: ASA congruence criterion

In \(\triangle ABD\) and \(\triangle CBD\), we have \(\angle ABD\cong\angle CBD\), \(BD = BD\), and \(\angle BDA\cong\angle BDC\). So, by the Angle - Side - Angle (ASA) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Answer:

1. Statements: \(\angle ABD\cong\angle CBD\); Reasons: Definition of angle bisector.
2. Statements: \(BD = BD\); Reasons: Reflexive property.
3. Statements: \(\triangle ABD\cong\triangle CBD\); Reasons: ASA (Angle - Side - Angle)