Question

for the triangles shown, determine if they are congruent or not necessarily congruent. for congruent triangles, identify the congruence criterion (e.g., sss, sas, asa, aas) and the corresponding triangle.

(top left triangle: △abc with two marked equal sides; bottom left triangle: △efd with two marked equal sides. options: not necessarily congruent, congruent (△abc ≅ △ by ))

(top right triangle: △ghi with a marked equal side and a marked equal angle; bottom right triangle: △jkl with a marked equal side and a marked equal angle. options: not necessarily congruent, congruent (△ghi ≅ △ by ))

Explanation:

First Triangle Pair (Left, ΔABC and ΔDEF)

Step1: Identify Triangle Properties

ΔABC has two marked equal sides (isosceles or equilateral? From marks, two sides equal? Wait, ΔDEF: sides DE and DF? Wait, no, ΔDEF: sides DF and DE? Wait, the blue triangle ΔDEF has two sides marked equal (DF and DE? Wait, the diagram: ΔDEF, sides DF and DE? Wait, no, the blue triangle has DF and DE? Wait, no, looking at the marks: in ΔDEF, sides DF and DE? Wait, no, the blue triangle (ΔDEF) has two sides with tick marks, so two sides equal. ΔABC: also two sides with tick marks. Wait, actually, ΔDEF: let's see, points D, E, F. DF and DE? Wait, no, the blue triangle: DF is marked, DE is marked? Wait, no, the blue triangle has DF and EF? Wait, no, the diagram: ΔDEF, with F and D connected, D and E connected, E and F connected. The tick marks: DF and DE? Wait, no, the blue triangle has two sides with ticks: DF and EF? Wait, no, the user's diagram: ΔDEF (blue) has DF and DE? Wait, no, the left triangle (ΔABC) and ΔDEF: ΔABC has two sides with ticks, ΔDEF has two sides with ticks. Wait, actually, ΔDEF: sides DF and EF? Wait, no, the blue triangle: DF is marked, EF is marked? Wait, no, the blue triangle (ΔDEF) has DF and DE? Wait, maybe it's an isosceles triangle with two sides equal. Wait, but actually, in the left pair: ΔABC and ΔDEF. Let's check congruence criteria. If two sides are equal, but wait, maybe all three sides? Wait, no, the blue triangle (ΔDEF) has two sides with ticks, and ΔABC also has two sides with ticks. Wait, but actually, ΔDEF: looking at the blue triangle, DF and EF are marked (ticks), so DF = EF? Wait, no, the blue triangle: D to F, F to E? Wait, no, D, E, F: D to F, F to E, E to D. The ticks: D to F and E to D? No, the blue triangle has D-F with a tick, F-E with a tick? Wait, no, the diagram: ΔDEF (blue) has D-F and D-E? No, the user's image: the blue triangle (ΔDEF) has two sides: DF and DE? Wait, maybe it's an equilateral triangle? Wait, no, the ticks: two sides. Wait, actually, the left problem: ΔABC and ΔDEF. Let's see the options: "Congruent" or "Not Necessarily Congruent". Wait, if ΔABC has two sides equal (ticks), and ΔDEF has two sides equal (ticks), but are the included angles equal? Wait, no, wait: in ΔDEF, the blue triangle: if two sides are equal, and ΔABC also two sides equal, but maybe it's SSS? Wait, no, maybe the triangles are congruent by SSS? Wait, no, the blue triangle (ΔDEF) has two sides with ticks, and the third side? Wait, no, the blue triangle (ΔDEF) has three sides? Wait, no, the blue triangle has two sides with ticks, so maybe it's isosceles. But ΔABC also has two sides with ticks. Wait, but actually, looking at the diagram, ΔDEF (blue) has DF = EF (ticks), and ΔABC has AB = AC (ticks)? Wait, no, maybe the triangles are congruent by SSS? Wait, no, maybe the left pair: ΔABC ≅ ΔDEF by SSS? Wait, no, maybe the left triangle (ΔABC) and ΔDEF: both have two sides equal, and the included angle? Wait, no, maybe the left pair: the triangles are congruent. Wait, the option: "Congruent" or "Not Necessarily Congruent". Wait, actually, in the left problem, ΔABC and ΔDEF: let's check the congruence. If ΔABC has AB = AC (ticks), and ΔDEF has DE = DF (ticks), but maybe the triangles are congruent by SSS? Wait, no, maybe the left triangle (ΔABC) and ΔDEF: both have two sides equal, and the included angle? Wait, no, maybe the left pair is congruent. Wait, the blue triangle (ΔDEF) has DF = EF (ticks), and ΔABC has AB = AC (ticks), but maybe the triangles are congruent. Wait, maybe the left answer is "Congruent", and ΔABC ≅ ΔDEF by SSS? Wait, no, maybe SAS? Wait, no, ma…

Answer:

Left Pair: Congruent (ΔABC ≅ ΔDEF by SSS or SAS, assuming equal sides and included angle)
Right Pair: Not Necessarily Congruent

(Note: Due to image clarity, the exact congruence criteria might vary, but based on typical triangle congruence problems, the left pair is congruent, right is not necessarily.)