Question

△ceg and △def are shown below.
which statement is true?
△ceg is similar to △def.
△ceg is not similar to △def.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Find angle at D in ΔDEF

In ΔDEF, we know two angles: ∠E = 64° and ∠D = 64°. The sum of angles in a triangle is 180°, so ∠F = 180° - 64° - 64° = 52°.

Step2: Find angle at C in ΔCEG

In ΔCEG, we know ∠E = 64° (wait, no, ∠G = 54°, ∠E: let's see, EF and FG are on the same line, so ∠E in ΔCEG: wait, actually, ∠E in ΔDEF and ∠E in ΔCEG? Wait, no, ΔCEG: points C, E, G. ∠G = 54°, ∠E: let's calculate ∠C. Sum of angles in triangle is 180°, so ∠C = 180° - ∠E - ∠G. Wait, but in ΔDEF, ∠F is 52°, and in ΔCEG, what's ∠E? Wait, maybe I made a mistake. Wait, ΔDEF: angles at E=64°, D=64°, so F=52°. ΔCEG: angle at G is 54°, angle at E: let's see, DE is a segment on CE, so ∠E in ΔCEG is the same as ∠E in ΔDEF? Wait, no, E is a common vertex? Wait, the diagram: E---F---G on a line, D on CE, F on EG? Wait, no, E, F, G are colinear. So ΔDEF has vertices D, E, F; ΔCEG has vertices C, E, G. So ∠E is common? Wait, no, ∠E in ΔDEF is at E, between D and F; ∠E in ΔCEG is at E, between C and G. Wait, maybe the key is to check if two angles are equal. In ΔDEF, angles are 64°, 64°, 52°. In ΔCEG, angle at G is 54°, angle at E: let's calculate angle at C. Wait, maybe I messed up. Wait, let's re-express:

In ΔDEF: ∠E = 64°, ∠D = 64°, so ∠F = 180 - 64 - 64 = 52°.

In ΔCEG: we need to find its angles. Let's see, ∠G = 54°, and since E, F, G are colinear, ∠E in ΔCEG: wait, maybe the triangles have a pair of equal angles? Wait, no, ΔDEF has angles 64, 64, 52; ΔCEG has angle at G=54, so let's find angle at C: 180 - ∠E - 54. But what's ∠E? Wait, maybe the problem is that in ΔDEF, the angles are 64, 64, 52, and in ΔCEG, angle at G is 54, so the angles don't match. Wait, no, maybe I made a mistake. Wait, the correct approach: for similarity, two angles must be equal (AA criterion). So in ΔDEF, angles are 64°, 64°, 52°. In ΔCEG, let's find its angles. Let's assume that ∠E in ΔCEG is equal to ∠E in ΔDEF (64°), then ∠C = 180 - 64 - 54 = 62°. So ΔDEF has angles 64, 64, 52; ΔCEG has 64, 54, 62. No two angles are equal, so they are not similar. Wait, but wait, maybe I misread the diagram. Wait, the problem: ΔCEG and ΔDEF. Let's check the angles again.

Wait, in ΔDEF: angles at D=64°, E=64°, so F=52°. In ΔCEG: angle at G=54°, angle at E: let's see, DE is part of CE, so ∠E in ΔCEG is the same as ∠E in ΔDEF (64°), so ∠C = 180 - 64 - 54 = 62°. So ΔDEF has angles 64, 64, 52; ΔCEG has 64, 54, 62. So no two angles are equal, so they are not similar. Therefore, the correct statement is "ΔCEG is not similar to ΔDEF."

Answer:

ΔCEG is not similar to ΔDEF.