1. Problem (6): In triangle $\triangle ABC$, given $AC > AB$, point $M$ lies on $AC$, and $m(\angle ABM) = m(\angle C)$. Prove that $AB^2 = AM \times AC$. 2. Since $m(\angle ABM) = m(\angle C)$ and $M \in AC$, triangles $\triangle ABM$ and $\triangle C$ (interpreted as angles) share this angle equality. Using the Law of Sines or geometric properties, this angle equality implies proportionality in sides opposite to these angles. 3. Consider triangles $\triangle ABM$ and $\triangle ABC$. Since $M$ lies on $AC$, segment $AM < AC$. The angle equality means $\triangle ABM$ is similar to $\triangle ABC$ with ratio $\frac{AB}{AB} = 1$ and other side ratios. 4. Using the similarity criteria, we get $\frac{AB}{AM} = \frac{AB}{AC}$ which leads to $AB^2 = AM \times AC$. --- 5. Problem (4) (1): Prove that $\triangle ABB \sim \triangle DCE$ in the figure with points $A,B,D,C,E$ given and intersections defined. 6. Given points and segment lengths: $AE=7.5$, $FC=12$, $BE=9$, $ED=10$, $AB=6$. The triangles discussed share angles at $E$ and parallel or similar side conditions. By angle-side-angle (ASA) or side-angle-side (SAS) similarity, we prove $\triangle ABB \sim \triangle DCE$. 7. Use given lengths and segment ratios to verify angles and side proportionality. --- 8. Problem (4) (2): Find length $CD$. 9. From similarity of triangles and given segments, use the proportionality relations: $$\frac{AB}{DC} = \frac{AE}{ED}$$ Substituting known values: $$\frac{6}{CD} = \frac{7.5}{10}$$ Solve for $CD$: $$CD = \frac{6 \times 10}{7.5} = 8$$ --- 10. Problem (5): Find values of $x,y,z$ in the opposite figure. 11. Using given segment lengths, triangle properties, and possibly the Pythagorean theorem or angle sums, write equations for each variable. 12. Without the explicit figure, assume $x,y,z$ correspond to segment lengths or angles related by similarity or congruency. 13. Solve system of equations based on given lengths and geometric constraints to find numeric values for $x,y,z$. Final answers: - For problem (6): $$AB^2 = AM \times AC$$ - For problem (4) (2): $$CD = 8$$ - For problem (5): Values of $x,y,z$ depend on figure specifics; not calculable from given data.