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Problem 121

# Given: $\underline{C A} \cong \underline{D B} \cdot \underline{C B} \cong \underline{D A} .$$Prove$$\triangle A B C \cong \triangle B A D$.

Expert verified
In the given problem, we have $$CA \cong DB$$ and $$CB \cong DA$$. Since AB is a common side of both triangles, we have $$AB \cong AB$$. By the Side-Side-Side (SSS) congruence theorem, we can conclude that $$\triangle ABC \cong \triangle BAD$$.
See the step by step solution

## Step 1: Identify congruent sides in the triangles

In the given problem, we have the following congruent sides: - Line segment CA is congruent to line segment DB. - Line segment CB is congruent to line segment DA. We can write these congruent relationships as $$CA \cong DB$$ and $$CB \cong DA$$.

## Step 2: Determine the third pair of congruent sides

From the given information, we can conclude that the third pair of congruent sides is AB, which is the common side of both triangles. We can write this as $$AB \cong AB$$.

## Step 3: Prove the triangles' congruence using SSS theorem

Now that we have all pairs of congruent sides, we can use the SSS congruence theorem to prove that the triangles are congruent. The SSS theorem states that if all three sides of one triangle are congruent to the three corresponding sides of another triangle, then the two triangles are congruent. We have: - $$CA \cong DB$$ - $$CB \cong DA$$ - $$AB \cong AB$$ By the SSS theorem, we can conclude that $$\triangle ABC \cong \triangle BAD$$.

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