Question

Complete.

3. You want to prove $△ABC\cong △XYZ$. You have shown localid="1649420823122" $\overline{\mathit{A}\mathit{B}}\cong \overline{XY}$and localid="1649420847157" $\overline{AC}\cong \overline{\mathit{X}\mathit{Z}}$. To prove the triangles congruent by SAS you must show that localid="1649420928994" $\underset{\_}{?}\cong \underset{\_}{?}$. To prove the triangles are congruent by SSS you must show that localid="1649420952959" $\underset{\_}{?}\cong \underset{\_}{?}$.

Short answer

The complete statement is: To prove the triangles congruent by SAS you must show that $\angle \mathit{A}\cong \angle \mathit{X}$and To prove the triangles congruent by SSS you must show that $\overline{BC}\cong \overline{YZ}$.

Step-by-step solution

- Write the SAS postulate and SSS postulate.

The SAS postulate states that if the two sides and the included angle of the one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

The SSS postulate states that if the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Step 2. Description of step.

It is being given that $△ABC\cong △XYZ$, $\overline{AB}\cong \overline{XY}$and$\overline{AC}\cong \overline{XZ}$.

As, the SAS postulate states that if the two sides and the included angle of the one triangle is congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Therefore, as, $\overline{AB}\cong \overline{XY}$and $\overline{AC}\cong \overline{XZ}$, thereforeto prove the triangles congruent by SAS, it must be proved that the included angle between the sides $\overline{AB}$and $\overline{AC}$in the triangle $△ABC$ andthe included angle between the sides $\overline{XY}$and $\overline{XZ}$in the triangle $△XYZ$must be congruent.

Therefore, $\angle A\cong \angle X$.

As, the SSS postulate states that if the three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Therefore, as, $\overline{AB}\cong \overline{XY}$and $\overline{AC}\cong \overline{XZ}$, thereforeto prove the triangles congruent by SSS, it must be proved that the third sides of the triangles $△ABC$and $△XYZ$must be congruent.

Therefore, $\overline{BC}\cong \overline{YZ}$.

Step 3. Write the complete statement.

Therefore, the complete statement is: To prove the triangles congruent by SAS you must show that $\angle A\cong \angle X$and To prove the triangles congruent by SSS you must show that $\overline{BC}\cong \overline{YZ}$.