Chapter Closure
Sign In
Looking at the given triangles, we see that △ RTS and △ ABC have two congruent interior angles. This is enough to conclude the similarity of these triangles by the Angle-Angle Similarity Theorem. rl △ ABC ~ △ RTS & (AA ~) The order in which the letters are listed tells us which vertices correspond to each other. Because of that, we can tell that AB corresponds to RT. However, as these lengths are not equal, the triangles are not congruent.
Looking at the given triangles, we see that △ RTS and △ ABC have two congruent interior angles. This is enough to conclude the similarity of these triangles by the Angle-Angle Similarity Theorem. rl △ ABC ~ △ MPK & (AA ~) The order in which the letters are listed tells us which vertices correspond to each other. Because of that, we can tell that AB corresponds to MP. Since these lengths are equal, we can conclude that △ ABC and △ MPK are congruent by the Angle-Angle-Side Congruence Theorem. rl △ ABC ≅ △ MPK & (AAS ≅)
We see that ∠ BCA ≅ ∠ XYZ, CA = YZ, and AB = ZX. If we knew that BC = XY, then we could conclude that △ ABC ≅ XZY by the Side-Angle-Side Congruence Theorem. Since we do not know that, the triangles might be congruent, or even similar. Consider a circle Z passing through X and a ray YX.
We see that Z and YX have two points of intersection: point X and point X'. We also have ZX = ZX' as Z passes through both X and X'. We see that △ X'ZY satisfies the same conditions as △ XZY, but they are not congruent. Therefore, we cannot conclude if either of them is congruent, or similar, to △ ABC.