Question

triangles abc and dbc have the following characteristics:
\overline{bc} is a side of both triangles
\angle acb and \angle dcb are right angles
\overline{ac} \cong \overline{dc}
which congruence theorem can be used to prove \triangle abc \cong \triangle dbc?
\bigcirc aas
\bigcirc sss
\bigcirc hl
\bigcirc sas

Explanation:

Step1: Recall congruence theorems

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

Step2: Analyze given information

• We know that \(\angle ACB=\angle DCB = 90^{\circ}\) (right angles), so \(\triangle ABC\) and \(\triangle DBC\) are right - angled triangles.
• \(BC\) is a common side (hypotenuse for both right - angled triangles as it is opposite the right angle) and \(AC\cong DC\) (one leg of each right - angled triangle).

Since we have a right - angled triangle situation with the hypotenuse (\(BC\)) and a leg (\(AC\cong DC\)) congruent, the HL (Hypotenuse - Leg) congruence theorem can be used to prove \(\triangle ABC\cong\triangle DBC\).

Answer:

HL