Question

state the third congruence that must be given to prove that ( \triangle jrm cong \triangle dfb ) using the indicated postulate.
a) given: ( overline{jr} cong overline{df}, overline{jm} cong overline{db} ); ( overline{}cong overline{} ) by sss congruence postulate
b) given: ( overline{jr} cong overline{df}, overline{jm} cong overline{db} ); ( overline{}cong overline{} ) by sas congruence postulate
c) given: ( overline{rm} cong overline{fb}, angle j ) is a right angle, and ( angle j cong angle d ); ( overline{}cong overline{} ) by hl congruence theorem

Explanation:

a)

Step1: Recall SSS Congruence Postulate

SSS (Side - Side - Side) congruence postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
We are given \( \overline{JR}\cong\overline{DF}\) and \( \overline{JM}\cong\overline{DB}\). For \( \triangle JRM\cong\triangle DFB\) by SSS, we need the third pair of sides \( \overline{RM}\cong\overline{FB}\).

b)

Step1: Recall SAS Congruence Postulate

SAS (Side - Angle - Side) congruence postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
We are given \( \overline{JR}\cong\overline{DF}\) and \( \overline{JM}\cong\overline{DB}\). The included angle for \( \overline{JR}\) and \( \overline{JM}\) in \( \triangle JRM\) is \( \angle J\), and the included angle for \( \overline{DF}\) and \( \overline{DB}\) in \( \triangle DFB\) is \( \angle D\). So we need \( \angle J\cong\angle D\).

c)

Step1: Recall HL Congruence Theorem

HL (Hypotenuse - Leg) congruence theorem states that if the hypotenuse and a leg of a right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.
We are given \( \overline{RM}\cong\overline{FB}\) (hypotenuse) and \( \angle J\cong\angle D\) (right angles). For \( \triangle JRM\cong\triangle DFB\) (right - triangles) by HL, we need the legs \( \overline{JR}\cong\overline{DF}\).

Answer:

a) \( \overline{RM}\cong\overline{FB}\)
b) \( \angle J\cong\angle D\)
c) \( \overline{JR}\cong\overline{DF}\)