Question

consider the diagram. the congruence theorem that can be used to prove △bae ≅ △cad is
○ sss
○ asa
○ sas
○ hl

Explanation:

1. First, identify the given information from the diagram:

• \( \angle BAE \) and \( \angle CAD \) are right angles (since \( BA \perp EC \) at \( A \)), so \( \angle BAE=\angle CAD = 90^\circ \).
• \( AE = AC \) (marked with one tick mark, indicating they are congruent segments).
• \( BE = CD \) (marked with two tick marks, indicating they are congruent segments).

1. Now, recall the congruence theorems:

• HL (Hypotenuse - Leg) Theorem: This theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.
• In \( \triangle BAE \) and \( \triangle CAD \):
• Both are right triangles (right - angled at \( A \)).
• Hypotenuse \( BE = CD \) (given by the tick marks).
• Leg \( AE = AC \) (given by the tick marks).
• Let's analyze other options:
• SSS (Side - Side - Side): We only know two sides (one leg and hypotenuse) are congruent, not all three sides. So SSS is not applicable.
• ASA (Angle - Side - Angle): We know one angle (\( 90^\circ \)) and one side (\( AE = AC \)), but we don't have the included side or the other angle in the ASA format. So ASA is not applicable.
• SAS (Side - Angle - Side): The angle in SAS should be the included angle between the two sides. Here, the right angle is not the included angle between \( AE, BE \) or \( AC, CD \) in the SAS sense. So SAS is not applicable.
• Since we have a right triangle, and we know the hypotenuse and one leg are congruent, the HL theorem is applicable.

Answer:

HL (The congruence theorem is HL, so the correct option is the one labeled "HL")