Isometry of Triangles
Two triangles 
and 
are said to be isometric with respect to a pairing or correspondence
of their first, second and third vertices, sides and angles when and only when matched sides have equal lengths and matched angles are equal.
Shorthand: The expression
means triangle 
is isometric to 
via
the correspondence of their first, second and third vertices. The correspondence
usually matches the largest side and angle of one with the largest side and
angle in the other; smallest side and angle of one with the largest side
and angle in the other; and the remaining third side and angles - exception
occurs for isosceles and equilateral triangle in which some sides and angles are
equal within a single triangle.
Remark (excuse the repetition). When we write
we assume the correspondences
between the first, second and third vertices in each triangle yields the correspondence of equal sides and angles.
Triangle Construction and Isometry Criteria.
The side-side-side, side-angle-side and angle-side-angle triangle construction methods are described in the following pages. These methods (SSS, SAS, ASA) when they work lead to triangles that are isometric. Beyond that, they lead to criteria deciding when a pair of triangles are isometric with respect to some correspondence. See details in following webpages.
Food for thought:
- The side-side-side method fails when the length of the longest side exceeds the sum of the two other sides.
- The angle-side-angle method fails when the the sum of the given angles exceeds or equals two right angles (180 degrees).
- The side-angle-side method fails when the angle > a straight angle.
See details below in following webpages.
Euclidean Geometry
with a geometry based
based development of
complex numbers
24 Lessons:
Easy Consequences of this (newest) Complex Number. Starter Lesson in this site folder follow below.