Question

explain what additional information, if any, would you need to prove δabc ≅ δdec by each postulate or theorem. 3. sss 4. sas

Explanation:

Step1: Analyze SSS Postulate

To prove \(\triangle ABC \cong \triangle DEC\) by SSS (Side - Side - Side) postulate, we need three pairs of corresponding sides to be equal. From the diagram, we know that \(\angle ACB\) and \(\angle DCE\) are vertical angles, so \(\angle ACB=\angle DCE\), and we can see that \(AC = DC\) (marked with the same tick mark). For SSS, we already have one pair of sides (\(AC = DC\)). We need to show that \(AB=DE\) and \(BC = EC\) (or \(AB = DE\) and \(BC=EC\) to complete the three pairs of equal sides).

Step2: Analyze SAS Postulate

To prove \(\triangle ABC \cong \triangle DEC\) by SAS (Side - Angle - Side) postulate, we need two pairs of corresponding sides and the included angle to be equal. We know that \(AC = DC\) (marked) and \(\angle ACB=\angle DCE\) (vertical angles). So we need either \(BC = EC\) (to have \(BC = EC\), \(\angle ACB=\angle DCE\), \(AC = DC\)) or \(AB = DE\) (but wait, the included angle for \(AB\) and \(AC\) is \(\angle A\), and for \(DE\) and \(DC\) is \(\angle D\). Wait, no, the correct corresponding sides for SAS with the included angle \(\angle ACB=\angle DCE\) are \(BC\) and \(EC\) (with \(AC = DC\)). So if we have \(BC = EC\), then with \(AC = DC\) and \(\angle ACB=\angle DCE\), we can use SAS. Alternatively, if we know that \(\angle A=\angle D\) and \(AB = DE\), but that would be ASA or AAS. Wait, for SAS, the included angle is between the two sides. So for \(\triangle ABC\) and \(\triangle DEC\), the sides around \(\angle ACB\) are \(BC\) and \(AC\), and around \(\angle DCE\) are \(EC\) and \(DC\). Since \(AC = DC\), we need \(BC = EC\) to apply SAS.

Answer:

• For SSS: Need \(AB = DE\) and \(BC=EC\) (or show that the third pair of sides are equal, since \(AC = DC\) is known).
• For SAS: Need \(BC = EC\) (since \(AC = DC\) and \(\angle ACB=\angle DCE\) (vertical angles) are known, so adding \(BC = EC\) gives the two sides and included angle for SAS).