Question

For the following figure, (a) List two pairs of congruent corresponding sides and one pair of congruent corresponding angles in $\Delta YTR$and $\Delta XTR$. (b) Notice that, in each triangle, you listed two sides and nonincluded angle. Do you think that SSA is enough to guarantee that two triangles are congruent?

Short answer

a. The two pairs of congruent corresponding sides and angles are$\overline{YT}\cong \overline{XT},\overline{TR}\cong \overline{TR}$and$\angle R\cong \angle R$.

b.These triangles can never be congruent and SSA is not postulated.

Step-by-step solution

Part a. Step 1. Consider the figure.

Consider the figure.

Part a. Step 2. Apply the AAA, SSS, and ASA postulates.

SSS Postulate: if the sides of a triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of the triangle are equal to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of a triangle are equal to the two corresponding angles and the included sides of another triangle, then the triangles are congruent.

Part a. Step 3. Step description.

By SAS postulate, in $\Delta YTR$and $\Delta XTR$, the two pairs of congruent corresponding sides are, $\overline{YT}\cong \overline{XT}\text{and,}\overline{TR}\cong \overline{TR}$.

And, one pair of congruent angles is $\angle R\cong \angle R$.

Therefore, the two pairs of congruent corresponding sides are $\overline{YT}\cong \overline{XT}\text{and,}\overline{TR}\cong \overline{TR}$and the congruent angles is $\angle R\cong \angle R$.

Part b. Step 1. Consider the figure.

Consider the figure.

Part b. Step 2. Apply the AAA, SSS, and ASA postulates.

SSS Postulate: if the sides of a triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of the triangle are equal to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of a triangle are equal to the two corresponding angles and the included sides of another triangle, then the triangles are congruent.

Part b. Step 3. Step description.

No, SSA is not enough to guarantee that two triangles are congruent. As from the figure $\Delta XTR$is a part of $\Delta YTR$. Thus, these triangles can never be congruent and SSA is not postulated.

Therefore, these triangles can never be congruent and SSA is not postulated.