Question

a side - side - side congruent triangle theorem
b side - angle - side congruent triangle theorem
c angle - angle - side congruent triangle theorem
d angle - side - angle congruent triangle theorem
e hypotenuse leg congruent triangle theorem
f not congruent

Explanation:

Step1: Analyze the given information

We have two triangles \(\triangle ABC\) and \(\triangle DBC\). We know that \(AB = CD\) (marked with one - cross), \(\angle A=\angle D\) (given as angles at \(A\) and \(D\)), and \(AC\) is a common side.

Step2: Recall the congruent triangle theorems

• Side - Side - Side (SSS): \(a = a',b = b',c = c'\) (three sides of one triangle equal to three sides of another triangle)
• Side - Angle - Side (SAS): Two sides and the included angle of one triangle equal to two sides and the included angle of another triangle (\(a = a',\angle=\angle,b = b'\))
• Angle - Angle - Side (AAS): Two angles and a non - included side of one triangle equal to two angles and the corresponding non - included side of another triangle (\(\angle=\angle,\angle=\angle,a = a'\))
• Angle - Side - Angle (ASA): Two angles and the included side of one triangle equal to two angles and the included side of another triangle (\(\angle=\angle,a = a',\angle=\angle\))
• Hypotenuse - Leg (HL): For right - angled triangles, hypotenuse and one leg of one right - angled triangle equal to hypotenuse and one leg of another right - angled triangle

In our case, we have \(AB = CD\), \(\angle A=\angle D\), and \(AC\) is common. So we have two sides (\(AB = CD\) and \(AC=AC\)) and the included angle (\(\angle A=\angle D\))

Answer:

B. Side - Angle - Side Congruent Triangle Theorem