Question

triangles abc and dbc have the following characteristics: bc is a side of both triangles ∠acb and ∠dcb are right angles (overline{ac}congoverline{dc}) which congruence theorem can be used to prove (\triangle abccong\triangle dbc)? o aas o sss o hl o 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.
• 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.
• We are given that \(AC\cong DC\) (a leg of each right - triangle) and \(BC\) is common to both triangles (the hypotenuse for both right - triangles as \(BC\) is opposite the right angles \(\angle ACB\) and \(\angle DCB\)).

Answer:

HL