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 \(\triangle FEC\)

• Vertical angles: \(\angle ACB=\angle FCE\) (vertical angles are equal).
• Given \(AC = EC\) (marked in the diagram).
• Given \(BC=FC\) (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, then the two triangles are congruent. Here, for \(\triangle ABC\) and \(\triangle FEC\), we have two sides (\(AC = EC\), \(BC = FC\)) and the included angle (\(\angle ACB=\angle FCE\)) equal.

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

• \(\angle I=\angle K = 90^{\circ}\) (right angles).
• \(HJ=HJ\) (common side).
• \(HI = HK\) (marked in the diagram).
• By the Hypotenuse - Leg (HL) congruence theorem: In a right - triangle, if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, then the two right - triangles are congruent. Here, \(HJ\) is the hypotenuse and \(HI\) and \(HK\) are legs for right - triangles \(\triangle HIJ\) and \(\triangle HKJ\) respectively.

Answer:

For \(\triangle ABC\) and \(\triangle FEC\):

• Enough Information: Yes
• Postulate or Theorem: SAS (Side - Angle - Side)

For \(\triangle HIJ\) and \(\triangle HKJ\):

• Enough Information: Yes
• Postulate or Theorem: HL (Hypotenuse - Leg)