STRAIGHT-LINE AND COMPASS CONSTRUCTIONS

Let us study about straight line and compass constructions,
You can draw straight lines, and set the compasses to any radius you like (obviously you can't measure the radius, because you've no markings on your ruler); in particular, if you've managed to mark two points on the (infinite) plane that you're drawing on (normally points are defined when arcs and lines intersect) then you can set the compass to exactly the distance between those points.

example : Draw an equilateral triangle.

Solution :
1. Draw a line, A.
2. Mark two arbitrary points on A, b and c, not coincident. ( The distance bc will be the length of each side of the triangle.)
3. Draw a circle, centre b, radius bc. ( This is two steps rolled into one: 1. Set the compasses to radius bc 2. Draw a circle centred on b )
4. Draw a circle, centre c, radius bc.
5. The two circles will intersect in two points, d and d', one either side of A.
6. Draw the line segments bd and cd. bcd is the equilateral triangle. ( Since lengths bd = bc = cd )

I hope the above explanation was useful.

Introduction to Similar Triangles

Let us understand what are similar triangles,
In general, to prove that two polygons are similar, you must show that all pairs of corresponding angles are equal and that all ratios of pairs of corresponding sides are equal. In triangles, though, this is not necessary.
Two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

Example 1: Use Figure 1 to show that the triangles are similar.

Figure 1 Similar triangles.

AA Similarity Postulate, Δ ABC ∼ Δ DEF. Additionally, because the triangles are now similar,

Hope the above explanation helped you.