In each row, the third figure comprises of a black circle and only those line segments which are not common to the first and the second figures.

1, 6, 9, are all triangles.

3, 4, 7 are all four-sided figures.

2, 5, 8 are all five-sided figures.

1, 2, 3 are figures composed of two straight lines.

4, 5, 6 are figures composed of three straight lines.

7, 8, 9 are figures composed of four straight lines.

1, 7, 8 are all undivided geometrical figures.

3, 5, 6 are geometrical figures divided into two parts.

2, 4, 9 are geometrical figures divided into four parts.

1, 3, 6 contain one complete circle each.

2, 5, 7 contain a semi-circle each. 

4, 8, 9 contain a triangle each.

1, 2, 7 are simple geometrical figures. 

3, 5, 9 have one element placed inside a different element.

4, 6, 8 have two different elements attached to each other.

The Horizontal lines are DF and BC i.e. 2 in number.

The Vertical lines are DG, AH and FI i.e. 3 in number.

The Slanting lines are AB, AC, BF and DC i.e. 4 in number.

Thus, there are 2 + 3 + 4 = 9 straight lines in the figure.

Now, we shall count the number of triangles in the figure.

The simplest triangles are ADE, AEF, DEK, EFK, DJK, FLK, DJB, FLC, BJG and LIC i.e. 10 in number.

The triangles composed of two components each are ADF, AFK, DFK, ADK, DKB, FCK, BKH, KHC, DGB and FIC i.e. 10 in number.

The triangles composed of three components each are DFJ and DFL i.e. 2 in number.

The triangles composed of four components each are ABK, ACK, BFI, CDG, DFB, DFC and BKC i.e. 7 in number.

The triangles composed of six components each are ABH, ACH, ABF, ACD, BFC and CDB i.e. 6 in number.

There is only one triangle i.e. ABC composed of twelve components.

There are 10 + 10 + 2 + 7 + 6+ 1 = 36 triangles in the figure.

When the triangles are drawn in an octagon with vertices same as those of the octagon and having one side common to that of the octagon, the figure will appear as shown in (Fig. 1).

Now, we shall first consider the triangles having only one side AB common with octagon ABCDEFGH and having vertices common with the octagon (See Fig. 2).Such triangles are ABD, ABE, ABF and ABG i.e. 4 in number.

Similarly, the triangles having only one side BC common with the octagon and also having vertices common with the octagon are BCE, BCF, BCG and BCH (as shown in Fig. 3). i.e. There are 4 such triangles.

This way, we have 4 triangles for each side of the octagon. Thus, there are 8 x 4 = 32 such triangles.