△abc and △dec are shown below.\nwhich statement is true?\n△abc is similar to △dec.\n△abc is not similar to…
△abc and △dec are shown below.\nwhich statement is true?\n△abc is similar to △dec.\n△abc is not similar to △dec.\nthere is not enough information to determine whether the triangles are similar.
Answer
Explanation:
Step1: Find angle at B in ΔABC
In triangle ( \triangle ABC ), the sum of angles is ( 180^\circ ). Given ( \angle A = 50^\circ ) and ( \angle ACB = 42^\circ ) (vertical angles with ( \angle DCE ), but here we use the angle at C in ( \triangle ABC )). So ( \angle B = 180^\circ - 50^\circ - 42^\circ = 88^\circ ).
Step2: Compare angles of the two triangles
In ( \triangle DEC ), we have ( \angle E = 88^\circ ), ( \angle DCE = 42^\circ ) (vertical angle with ( \angle ACB )), so ( \angle D = 180^\circ - 88^\circ - 42^\circ = 50^\circ ). Now, ( \triangle ABC ) has angles ( 50^\circ ), ( 42^\circ ), ( 88^\circ ) and ( \triangle DEC ) has angles ( 50^\circ ) ( ( \angle D ) ), ( 42^\circ ) ( ( \angle DCE ) ), ( 88^\circ ) ( ( \angle E ) ). By AA (Angle - Angle) similarity criterion, the triangles have two equal angles, so they are similar.