Question

In triangle CDE, if angle C is 30° and angle E is 50°, and in triangle FGH, angle G is 100° and angle H is 50°, are the triangles congruent?

1) Yes, triangle CDE ≅ triangle FGH.
2) Yes, triangle CDE ≅ triangle GHF.
3) Yes, triangle CDE ≅ triangle HGF.
4) No, the triangles are not necessarily congruent.

Answer 1

By calculating the third angle of each triangle and comparing them, we find that triangles CDE and FGH are similar due to having identical angles but not necessarily congruent as side measurements are unknown. Hence, the triangles are not necessarily congruent.

Step-by-step explanation:

To determine if the triangles CDE and FGH are congruent, we can use the property that the sum of the angles in any triangle is 180 degrees. For triangle CDE, we have angles C and E as 30° and 50° respectively, so the third angle D would be 180° - (30° + 50°) = 100°.

For triangle FGH, we have angles G and H as 100° and 50° respectively, so the third angle F is 180° - (100° + 50°) = 30°. Comparing the angles of both triangles:

Angle C of triangle CDE is 30° (same as angle F of FGH)

Angle E of triangle CDE is 50° (same as angle H of FGH)

Angle D is 100° (same as angle G of FGH)

With all corresponding angles equal, the triangles are similar by the Angle-Angle (AA) similarity criterion, but they are not necessarily congruent as we do not have information about the sides of the triangles to apply any congruence rules like Side-Angle-Side (SAS) or Side-Side-Side (SSS). Therefore, option 4 is correct: No, the triangles are not necessarily congruent.