Let's analyze and solve the problem based on the information given in the image.

The problem provides two congruent triangles, $\triangle ABC$ and $\triangle DEF$, with specific measurements and angles:

1. $\angle ABC = 95^\circ$
2. $DF = 12 \, \text{cm}$, $DE = 5 \, \text{cm}$
3. $\angle DEF = 50^\circ$

We are asked to find the values of:

• (a) $x$ (side $AC$ in $\triangle ABC$),
• (b) $y$ (angle $\angle BAC$ in $\triangle ABC$),
• (c) $z$ (angle $\angle DFE$ in $\triangle DEF$).

Since the triangles are congruent, corresponding sides and angles are equal. Let’s find each of the values:

(a) $x$

Since $\triangle ABC \cong \triangle DEF$, side $AC$ in $\triangle ABC$ is congruent to side $DE$ in $\triangle DEF$. Therefore: $x = DE = 5 \, \text{cm}$

(b) $y$

In $\triangle ABC$, we have: $\angle ABC = 95^\circ \quad \text{and} \quad \triangle ABC \cong \triangle DEF$ Since the sum of angles in a triangle is $180^\circ$, we can calculate $y$ (i.e., $\angle BAC$) by: $y = 180^\circ - 95^\circ - 50^\circ = 35^\circ$

(c) $z$

Since $\triangle ABC \cong \triangle DEF$, angle $z$ in $\triangle DEF$ corresponds to angle $y$ in $\triangle ABC$. Therefore: $z = y = 35^\circ$

Final Answers:

1. $x = 5 \, \text{cm}$
2. $y = 35^\circ$
3. $z = 35^\circ$

Would you like further details on these solutions, or do you have additional questions?

Here are some related questions for further practice:

1. How do we prove triangles are congruent using the Side-Angle-Side (SAS) criterion?
2. If the triangles were not congruent, how would that affect our answers?
3. Can you calculate the area of $\triangle DEF$ using the given information?
4. What are other methods to determine unknown angles in a triangle?
5. How would you solve for $x$, $y$, and $z$ if only one angle and two sides were known?

Tip: Remember, the sum of interior angles in any triangle is always $180^\circ$.