Question

5 finish the triangle proof by dragging the correct reasons to their box. there will be two extra unused reasons!
given: \\( \angle b a d \cong \angle d c b, b c \\| d a \\)
prove: \\( a b \cong c d \\)
\\( a b \cong c d \\) \\( b d \cong b d \\) \\( \triangle b a d \cong \triangle d c b \\) \\( \angle b a d \cong \angle d c b \\) asa corresponding angles reflexive property alternate interior angles given given aas

Explanation:

Step1: State the first given

Statement: \(BC\parallel DA\)
Reason: Given

Step2: Use parallel line property

Statement: \(\angle ADB\cong\angle CBD\)
Reason: Alternate Interior Angles (When two parallel lines \(BC\) and \(DA\) are cut by a transversal \(BD\), alternate - interior angles are congruent)

Step3: State the second given

Statement: \(\angle BAD\cong\angle DCB\)
Reason: Given

Step4: Use reflexive property

Statement: \(BD = BD\)
Reason: Reflexive Property (A segment is congruent to itself)

Step5: Prove triangle congruence

Statement: \(\triangle BAD\cong\triangle DCB\)
Reason: \(AAS\) (We have two pairs of congruent angles (\(\angle BAD\cong\angle DCB\) and \(\angle ADB\cong\angle CBD\)) and a pair of congruent non - included sides (\(BD = BD\)))

Step6: Use CPCTC

Statement: \(AB\cong CD\)
Reason: \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. \(BC\parallel DA\) - Given
2. \(\angle ADB\cong\angle CBD\) - Alternate Interior Angles
3. \(\angle BAD\cong\angle DCB\) - Given
4. \(BD = BD\) - Reflexive Property
5. \(\triangle BAD\cong\triangle DCB\) - \(AAS\)
6. \(AB\cong CD\) - \(CPCTC\)