Question

1. given: ( abcong ac ) and ( \triangle bdecong\triangle cde ) prove: ( angle badcongangle cad )

Explanation:

Step1: Use the given information

We are given \(AB\cong AC\) and \(\triangle BDE\cong\triangle CDE\). From \(\triangle BDE\cong\triangle CDE\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we get \(BD\cong CD\).

Step2: Consider \(\triangle ABD\) and \(\triangle ACD\)

We have \(AB\cong AC\) (given), \(BD\cong CD\) (from Step 1), and \(AD\cong AD\) (Reflexive Property of Congruence).

Step3: Prove \(\triangle ABD\cong\triangle ACD\)

By SSS (Side - Side - Side) congruence criterion, \(\triangle ABD\cong\triangle ACD\) since \(AB = AC\), \(BD=CD\), and \(AD = AD\).

Step4: Prove \(\angle BAD\cong\angle CAD\)

Since \(\triangle ABD\cong\triangle ACD\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle BAD\cong\angle CAD\).

Answer:

The statements and reasons are filled as follows:

Statement	Reason
\(AD\cong AD\)	Reflexive
\(BD\cong CD\)	CPCTC (\(\triangle BDE\cong\triangle CDE\))
\(\triangle ABD\cong\triangle ACD\)	SSS (\(AB = AC\), \(BD = CD\), \(AD=AD\))
\(\angle BAD\cong\angle CAD\)	CPCTC (\(\triangle ABD\cong\triangle ACD\))