1. **Problem:** Determine if the given pairs of triangles are similar, write the similarity statement, and name the theorem used. 2. **Key Theorems for Triangle Similarity:** - **AA (Angle-Angle) Similarity:** Two triangles are similar if two angles of one triangle are congruent to two angles of the other. - **SSS (Side-Side-Side) Similarity:** Two triangles are similar if their corresponding sides are in proportion. - **SAS (Side-Angle-Side) Similarity:** Two triangles are similar if two sides are in proportion and the included angle is congruent. --- ### a. - Given: Triangle RST with sides RT=10, ST=8; Triangle XYZ with angle X=20°, side XY=15. - We have side lengths for RST but only one angle and one side for XYZ. - Not enough information to confirm similarity. ### b. - Given: Triangle FGH with segment HJ; Triangle JKL with right angle at K. - Without more angle or side info, similarity cannot be established. ### c. - Triangle DEF: DE=4.5, EF=3, height from F=3. - Triangle ABC: AB=2, BC=3, AC=4. - Check side ratios: $$\frac{DE}{AB} = \frac{4.5}{2} = 2.25$$ $$\frac{EF}{BC} = \frac{3}{3} = 1$$ - Sides are not proportional, so no similarity by SSS. ### d. - Triangle MWR right angle at W, side WR=10. - Triangle GHE with sides HE=4, GH=3. - No angle info for GHE; cannot confirm similarity. ### e. - Triangle BXC: BX=25, XC=30. - Triangle AXY: AX=20, AY=25. - Check if triangles share angle X and if sides around X are proportional. - If angle X is common, and sides around X are proportional: $$\frac{BX}{AX} = \frac{25}{20} = 1.25$$ $$\frac{XC}{AY} = \frac{30}{25} = 1.2$$ - Ratios differ, so no similarity by SAS. ### f. - Triangle JKL: JK=32, KL=45, angle K=60°. - Triangle PSR: PS=40, SR=22, angle P=30°. - Angles differ, sides not proportional; no similarity. --- **Final conclusion:** None of the pairs have sufficient information or proportional sides/angles to confirm similarity. --- **Similarity statement:** None can be stated with given data. **Theorem used:** Not applicable due to insufficient data.