Given that line segment DE is parallel to line segment AB, determine how triangles DEC and ABC can be shown to be similar.
A. Since DCE ACB and DC = EC, the triangles are similar by angle-side
B. Since DCE ACB and AC = BC, the triangles are similar by angle-side.
C. Since DCE ACB and DEC ABC, the triangles are similar by angle-angle.
D. Since DCE ACB and DEC BAC, the triangles are similar by angle-angle.
C. Since ∠DCE ≅ ∠ACB and ∠DEC ≅ ∠ABC, the triangles are similar by angle-angle.
The correct answer is C. Since DCE and ACB have corresponding angles that are equal (the included angle between the parallel lines), and DEC and ABC have corresponding angles that are equal (the included angle between the parallel lines), the triangles can be shown to be similar by angle-angle.
To determine how triangles DEC and ABC can be shown to be similar, we need to consider the properties of similar triangles. Similar triangles have equal corresponding angles and proportional corresponding sides.
Given that line segment DE is parallel to line segment AB, we can determine the relationship between the angles and sides of triangles DEC and ABC using the Angle-Angle (AA) similarity criterion.
In the options provided:
A. Since DCE ACB and DC = EC, the triangles are similar by angle-side.
This option mentions the side DC = EC, which indicates a corresponding side ratio, but it does not provide information about corresponding angles. Therefore, this option is not sufficient to conclude similarity.
B. Since DCE ACB and AC = BC, the triangles are similar by angle-side.
This option also mentions the side AC = BC, which indicates a corresponding side ratio, but it does not provide information about corresponding angles. Therefore, this option is not sufficient to conclude similarity.
C. Since DCE ACB and DEC ABC, the triangles are similar by angle-angle.
This option states that angles DCE, ACB, and DEC, ABC are corresponding angles, which satisfies the Angle-Angle (AA) similarity criterion. Therefore, triangles DEC and ABC can be shown to be similar by the angle-angle condition.
D. Since DCE ACB and DEC BAC, the triangles are similar by angle-angle.
This option states that angles DCE, ACB, and DEC, BAC are corresponding angles, which does not satisfy the Angle-Angle (AA) similarity criterion. Therefore, this option is not valid for concluding similarity.
Therefore, the correct answer is option C: Since DCE ACB and DEC ABC, the triangles are similar by angle-angle.