Question

decided whether enough information is given to prove that the triangles are congruent. if there is enough information, state the congruence postulate of theorem you would use.

Explanation:

Step1: Analyze \(\triangle ABC\) and \(\triangle FEC\)

• We know that \(\angle ACB=\angle FCE\) (vertical angles are equal).
• We are given \(AC = FC\) and \(BC = EC\) (marked in the diagram).
• By the Side - Angle - Side (SAS) congruence postulate, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. For \(\triangle ABC\) and \(\triangle FEC\), \(AC = FC\), \(\angle ACB=\angle FCE\), \(BC = EC\), so there is enough information.

Step2: Analyze \(\triangle HIJ\) and \(\triangle HKJ\)

• We know that \(\angle HJI=\angle HKJ = 90^{\circ}\) (right angles).
• \(HJ=HJ\) (common side).
• \(HI = HK\) (marked in the diagram).
• By the Hypotenuse - Leg (HL) congruence theorem (for right - angled triangles, if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, the triangles are congruent). Here, \(HJ\) is the hypotenuse (common) and \(HI = HK\) (legs), so there is enough information.

Answer:

• For \(\triangle ABC\) and \(\triangle FEC\): Yes, SAS (Side - Angle - Side)
• For \(\triangle HIJ\) and \(\triangle HKJ\): Yes, HL (Hypotenuse - Leg)