Question

if δcat ≅ δdog, determine which of parts (a) through (c), if any, are necessarily true. explain why.
a. is (overline{ct}congoverline{dg}) necessarily true?
a. yes. since the triangles are congruent and (overline{ct}) corresponds to (overline{dg}), (overline{ct}congoverline{dg})
b. no. (overline{ct}) and (overline{dg}) are not corresponding line segments in the two triangles.
b. is (angle acongangle d) necessarily true?
a. yes. since the triangles are congruent and (angle a) corresponds to (angle d), (angle acongangle d).
b. no. (angle a) and (angle d) are not corresponding angles in the triangles.
c. is (overline{at}congoverline{do}) necessarily true?
a. yes. since the triangles are congruent and (overline{at}) corresponds to (overline{do}), (overline{at}congoverline{do})
b. no. (overline{at}) and (overline{do}) are not corresponding line segments in the two triangles.

Explanation:

Step1: Recall congruent - triangle property

When two triangles $\triangle CAT\cong\triangle DOG$, corresponding parts are congruent.

Step2: Analyze part (a)

The vertices of $\triangle CAT$ and $\triangle DOG$ are written in corresponding order. $C$ corresponds to $D$, $A$ corresponds to $O$, and $T$ corresponds to $G$. So, $\overline{CT}$ corresponds to $\overline{DG}$. Since the triangles are congruent, corresponding sides are congruent. So, $\overline{CT}\cong\overline{DG}$.

Step3: Analyze part (b)

In $\triangle CAT\cong\triangle DOG$, $A$ corresponds to $O$, not $D$. So, $\angle A$ and $\angle D$ are not corresponding angles, and $\angle A
ot\cong\angle D$.

Step4: Analyze part (c)

In $\triangle CAT\cong\triangle DOG$, $\overline{AT}$ corresponds to $\overline{OG}$, not $\overline{DO}$. So, $\overline{AT}$ and $\overline{DO}$ are not corresponding line - segments, and $\overline{AT}
ot\cong\overline{DO}$.

Answer:

a. A. Yes. Since the triangles are congruent and $\overline{CT}$ corresponds to $\overline{DG}$, $\overline{CT}\cong\overline{DG}$
b. B. No. $\angle A$ and $\angle D$ are not corresponding angles in the triangles
c. B. No. $\overline{AT}$ and $\overline{DO}$ are not corresponding line segments in the two triangles