Question

point m and n are midpoints of ab and da respectively. complete the following table to prove \\( \triangle cdn \cong \triangle cbm \\) for the last reason choose from: sss, sas, or asa.

Explanation:

Step1: Identify congruent sides

Since it is a square, \(DC\cong BC\) (the sides of a square are congruent).

Step2: Identify other congruent sides

Point \(M\) is the mid - point of \(AB\) and point \(N\) is the mid - point of \(DA\). Let the side length of the square be \(s\). Then \(BM=\frac{1}{2}AB\) and \(DN = \frac{1}{2}DA\). Since \(AB = DA\) (sides of a square), \(BM\cong DN\).

Step3: Identify congruent angles

In a square, \(\angle D\cong\angle B\) (all angles of a square are right angles, so they are congruent).

Step4: Determine the congruence criterion

We have two sides and the included angle congruent. For \(\triangle CDN\) and \(\triangle CBM\), we have \(DC = BC\), \(DN=BM\), \(\angle D=\angle B\). By the Side - Angle - Side (SAS) criterion, \(\triangle CDN\cong\triangle CBM\).

Answer:

Statement	Reason
\(\overline{BM}\cong\overline{DN}\)	They are both half the length of a side of the square.
\(\angle D\cong\angle B\)	All the angles of a square are congruent.
\(\triangle CDN\cong\triangle CBM\)	SAS