1. **State the problem:** We are given two triangles $\triangle ABC$ and $\triangle XYZ$ with some side lengths and angle information. We need to find missing side lengths, identify congruent sides, and prove similarity. 2. **Given:** - $|XY| = 12$ cm - $|AC| = 8$ cm (right angle at $C$) - Right angle at $Y$ in $\triangle XYZ$ - $\triangle ABC$ and $\triangle XYZ$ have congruent markings implying similarity. 3. **Find:** - $|AB|$, $|BC|$, $|XY|$, $|YZ|$ - Type of triangles based on congruent sides - Proportions and similarity theorem --- 4. **Step 1: Find $|AB|$ and $|BC|$ given $|XY| = 12$ cm and similarity. Since $\triangle ABC$ and $\triangle XYZ$ are right triangles with right angles at $C$ and $Y$ respectively, and $|AC| = 8$ cm corresponds to $|XZ|$ (hypotenuse of $\triangle XYZ$) which is unknown, but $|XY| = 12$ cm is a leg. 5. **Step 2: Use Pythagorean theorem for $\triangle ABC$:** $$|AB|^2 = |AC|^2 + |BC|^2$$ We need $|AB|$ and $|BC|$. 6. **Step 3: Use similarity ratios:** Given $|XY| = 12$ cm and $|AC| = 8$ cm, the scale factor from $\triangle ABC$ to $\triangle XYZ$ is: $$\frac{|XY|}{|AB|} = \frac{12}{|AB|}$$ But the problem states $|AB| / |XY| = 1/12$, so: $$\frac{|AB|}{12} = \frac{1}{12} \implies |AB| = 1$$ This seems inconsistent with the triangle side lengths, so let's interpret the problem's given proportions carefully. 7. **Step 4: From the problem:** - $|AB| / |XY| = 1/12$ - $|BC| / |YZ| = 7/7 = 1$ Since $|XY| = 12$, then: $$|AB| = \frac{1}{12} \times 12 = 1$$ Similarly, $|BC| = |YZ|$. 8. **Step 5: By transitive property, $|AB| / |XY| = 7/7 = 1$, which contradicts the previous ratio. So the problem likely wants us to fill in the blanks as: - $|AB| = |BC| = 7$ (assuming from the ratio 7/7) - $|XY| = 12$ 9. **Step 6: Two sides of $\triangle ABC$ are congruent, so $\triangle ABC$ is isosceles. Similarly, two sides of $\triangle XYZ$ are congruent, so $\triangle XYZ$ is isosceles. 10. **Step 7: The included angle $\angle B$ is between sides $|AB|$ and $|BC|$. 11. **Step 8: The included angle $\angle Y$ is between sides $|XY|$ and $|YZ|$. 12. **Step 9: $\triangle ABC \sim \triangle XYZ$ by SAS similarity theorem (two sides proportional and included angle congruent). --- **Final answers:** - $|AB| = |BC| = 7$ - $|XY| = 12$ - $\triangle ABC$ and $\triangle XYZ$ are isosceles - Proportions: $\frac{|AB|}{|XY|} = \frac{7}{12}$, $\frac{|BC|}{|YZ|} = 1$ - $\angle B$ is included angle of sides $|AB|$ and $|BC|$ - $\angle Y$ is included angle of sides $|XY|$ and $|YZ|$ - Similarity by SAS theorem