Which of the following figures represent congruent triangles?

To determine which figures represent congruent triangles, we need to examine the markings on each figure. Congruent triangles have corresponding sides and angles that are equal.

• Figure 1: The diagonals intersect. We see markings indicating that the segments of the diagonals are equal (AD = DB and AC = CB). This implies that the triangles might be congruent, but without information about angles or more sides, we cannot definitively conclude congruence based on standard triangle congruence postulates (SSS, SAS, ASA, AAS, HL).
• Figure 2: This figure shows two triangles (MPK and NKP) formed by intersecting lines. The markings indicate that MP = NK and PK = MK. Also, angles at K are marked with arcs, suggesting they might be equal, or possibly a straight angle if M, K, N are collinear. However, the markings on the sides (MP=NK, PK=MK) and the angles at P and N being marked with arcs suggest we might be looking for congruence criteria. If we assume angle MPK = angle NKP and angle PMK = angle KNP, then we might have congruence. However, the standard markings of two arcs at P and N suggest these angles are equal. With two pairs of equal sides (MP=NK, PK=MK) and one pair of equal angles (angle P = angle N), this would not be enough for congruence unless the angles were included between the sides.
• Figure 3: This figure shows a quadrilateral ABCD with a diagonal AC. The markings indicate AD = DC and angle DAC = angle DCA. This means triangle ADC is an isosceles triangle. The markings on AB and BC (single tick marks) indicate AB = BC. With two pairs of equal sides (AD = DC, AB = BC) and one pair of equal angles (angle DAC = angle DCA), this is not sufficient for congruence. If the angle BAC = angle BCA were also marked, then we would have SAS (if angle A = angle C was marked) or ASA/AAS.
• Figure 4: This figure shows a quadrilateral ABCD with a diagonal BD. The markings indicate AD = BC (single tick marks) and angle ADB = angle CBD (arcs). The diagonal BD is common to both triangles ABD and CDB. However, there's no clear indication of congruence here. If AB = CD was marked, we would have SAS or SSS. If angle DAB = angle BCD was marked, we might have other cases.
• Figure 5: This figure shows a triangle MDF and a triangle EDF. The angle at F is marked with a full circle and a line, indicating a straight angle (180 degrees). Angles MFD and EFD are marked with arcs. There are also markings on MD and ED, and MF and EF. The angle marked at D suggests angle MDF = angle EDF. There are double tick marks on MF and EF (MF=EF), and single tick marks on MD and ED (MD=ED). The common side FD. So, triangle MDF and triangle EDF have three pairs of equal sides (MF=EF, MD=ED, FD=FD). Therefore, by SSS (Side-Side-Side) congruence postulate, triangle MDF is congruent to triangle EDF.
• Figure 7: This figure shows a quadrilateral MKNP with a diagonal KN. Markings indicate MK = NP (double tick marks) and MN = KP (single tick marks). The diagonal KN is common to both triangles MKN and NKP. So, we have two pairs of equal sides (MK = NP, MN = KP) and a common side (KN = NK). Therefore, by SSS (Side-Side-Side) congruence postulate, triangle MKN is congruent to triangle NKP.

Based on the markings, figures 5 and 7 clearly show congruent triangles by the SSS postulate.

Answer: Figures 5 and 7 represent congruent triangles.