Triangle Similarity That Involves Special Segments, and Perimeters in Triangles. Angle Bisector Theorem.

We have learnt that similar triangles have corresponding sides that are proportional. This statement may be extended to include medians, altitudes, angle bisectors, and perimeters proportional with these sides.

You will go over a lesson that starts by reviewing similarity highlighting the sides that are corresponding, and setting up extended proportions. Once the review is completed, the lesson shifts to solve problems with special segments. The lesson focuses in the reading process of the text in the examples. You will be given a few problems to solve after each example. Go and try the lesson!

Lesson's Content

Lesson's Glossary

Altitude
Height

Altitude of a triangle
The perpendicular segment from a vertex to the line containing the opposite side of a triangle.

Angle bisector
A ray that is in the interior of an angle and forms two equal angles with the sides of that angle.

Angle-angle-angle (AAA) similarity
The angle-angle-angle (AAA) similarity test states that given two triangles that have corresponding angles that are congruent, then the triangles are similar. As we know the sum of the interior angles in a triangle is 180°, so if two corresponding are congruent, then the other ones should be as well.

Median
The segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it.

Perpendicular bisector
The bisector of a segment perpendicular to it.

Side-angle-side (SAS) similarity
The side-angle-side (SAS) similarity test states that given two triangles that have two pairs of sides that are proportional and the included angles are congruent, then the triangles should be similar.

Side-side-side (SSS) similarity
The side-side-side (SSS) similarity test states that for two triangles to be similar; all corresponding sides should be proportional.

Similar
two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.

Similar triangles
Similar triangles are triangles which have the same shape but probably different size. Corresponding angles need to be congruent, and corresponding sides are in proportion.

Triangle
A polygon with three sides.

Interactive Geometric Applets: Relevant Theorems.

Similar triangles have the same shape but different size. If you want to enlarge a

triangle; you need to multiply each side by a scale factor. In a reduction you divide

by the scale factor. The applet below displayed allows you to visualize this

similarity relationship.

Triangles have altitudes, medians, and angle bisectors.

You have these definitions in the applet below. You may drag any vertex to test the relationships

specified in the definitions.

Three perpendicular bisector in a triangle intersect at the circumcenter,

from the circumcenter you may draw a circle circumscribing the triangle.

The radius of this circle is called "circumradius", and the circle is also known

as circumcircle. Move the vertices in this construction to verify these relationships.

Three angle bisectors in a triangle intersect at the incenter. From the incenter you may draw the incircle, or inscribed

circle, whose radius is the inradius. Drag any of the vertices to verify this relationship.

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