Question

problem 23
given: ( angle b cong angle d ) and ( angle a cong angle c )
prove: ( overline{ab} cong overline{dc} )

1. ( angle b cong angle d ) and ( angle a cong angle c )
2. given
3. click here to insert
4. click here to insert
5. click here to insert
6. click here to insert

Explanation:

Step1: Use the Angle - Angle - Side (AAS) Congruence Theorem

If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
In \(\triangle ABD\) and \(\triangle CDB\), we have \(\angle B\cong\angle D\), \(\angle A\cong\angle C\), and \(BD = BD\) (common side).
So, \(\triangle ABD\cong\triangle CDB\) (by AAS).

Step2: Use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Since \(\triangle ABD\cong\triangle CDB\), then the corresponding sides \(AB\) and \(DC\) are congruent.

Answer:

1. \(\triangle ABD\cong\triangle CDB\) ; Angle - Angle - Side (AAS) Congruence Theorem
2. \(\overline{AB}\cong\overline{DC}\) ; Corresponding Parts of Congruent Triangles are Congruent (CPCTC)