Question

given the shaded gray triangles below with sides and angles as marked.
select the postulate or theorem that proves these triangles are congruent.
angle angle side congruence theorem
these triangles are not congruent.
angle side angle congruence postulate
side side side congruence postulate
hypotenuse leg congruence theorem
side angle side congruence postulate

Explanation:

Step1: Recall congruence postulates and theorems

• Angle - Angle - Side (AAS): If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.
• Side - Side - Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
• Hypotenuse - Leg (HL): For right - triangles, if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.
• Angle - Side - Angle (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
• Side - Angle - Side (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Step2: Analyze the given triangles

Assume the two gray triangles are right - triangles (since they are part of a larger geometric figure which may imply right - angles). If we consider the non - vertical sides (legs) as equal (marked as equal in the figure) and the hypotenuses (the non - common sides) as equal (by the property of the larger figure, say an isosceles triangle split into parts), and the right - angles (implied by the geometric construction)
The Hypotenuse - Leg Congruence Theorem states that for two right - triangles, if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.

Answer:

Hypotenuse Leg Congruence Theorem