1. Problem: Prove that two triangles are congruent using the Side-Angle-Side (SAS) criterion. Formula: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. Explanation: This means if $AB = DE$, $\angle BAC = \angle EDF$, and $AC = DF$, then $\triangle ABC \cong \triangle DEF$. 2. Problem: Prove congruency of two right triangles using the Hypotenuse-Leg (HL) theorem. Formula: In right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another triangle, the triangles are congruent. Explanation: If $\triangle ABC$ and $\triangle DEF$ are right triangles with $AB = DE$ (hypotenuses) and $BC = EF$ (legs), then $\triangle ABC \cong \triangle DEF$. 3. Problem: Prove that the mid-segment of a triangle is parallel to the third side and half its length. Formula: The segment joining the midpoints of two sides of a triangle is parallel to the third side and equals half its length. Explanation: If $M$ and $N$ are midpoints of $AB$ and $AC$ in $\triangle ABC$, then $MN \parallel BC$ and $MN = \frac{1}{2} BC$. 4. Problem: Prove congruency of two triangles in 3D space using the Side-Side-Side (SSS) criterion. Formula: If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent. Explanation: For triangles $\triangle ABC$ and $\triangle DEF$, if $AB=DE$, $BC=EF$, and $AC=DF$, then $\triangle ABC \cong \triangle DEF$. 5. Problem: Prove the height theorem in a right triangle. Formula: The altitude to the hypotenuse creates two triangles similar to the original triangle and to each other. Explanation: If $AD$ is the altitude from right angle $A$ to hypotenuse $BC$ in $\triangle ABC$, then $\triangle ABD \sim \triangle ADC \sim \triangle ABC$. 6. Problem: Prove the leg theorem in a right triangle. Formula: Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of that leg on the hypotenuse. Explanation: For leg $AB$ and projection $BD$ on hypotenuse $BC$, $AB^2 = BD \times BC$. 7. Problem: Prove similarity of two pyramids with equal heights and proportional bases. Formula: Two pyramids are similar if their corresponding base areas are proportional and their heights are equal. Explanation: If pyramids $P$ and $Q$ have bases with areas $A_1$ and $A_2$ such that $\frac{A_1}{A_2} = k^2$ and equal heights, then $P \sim Q$ with scale factor $k$. 8. Problem: Prove area similarity between two triangles. Formula: The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides. Explanation: If $\triangle ABC \sim \triangle DEF$ with scale factor $k$, then $\frac{Area(ABC)}{Area(DEF)} = k^2$. Final answers are the congruency or similarity statements as shown.