Geometry: Similar Triangles

MA.912.G.4.5 Apply theorems involving segments divided

proportionallyBlock 28

Similar triangles

Definition: Two triangles are similar if and only if their

corresponding angles are congruent and corresponding sides are proportional.

Tests for similarity of triangles

(AA Similarity) If two angles of one triangles are congruent to two angles of another triangle, then the triangles are similar.

There is a lot of situations where similar triangles naturally arise

Tests for similarity of triangles cont.

• Side-Side-Side (SSS) Similarity: If the corresponding side lengths of two triangles are proportional, then triangles are similar

• SAS (SAS) Similarity: It the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent then the triangles are similar

Proofs using similar triangles

Segments of chords theorem

Theorem: Given AB and CD chords of a circle intersect at point F then AF*FB=CF*FD.

Two-column proof:

• Refer to the following picture:

Two-column proof:Statements:• AB and CD intersect at F• Angle A is congruent to

angle D

• Triangles AFC and DFB are similar

• AF/FD=CF/FB

• AF*FB=CF*FD

Reasoning:• Given• Inscribed angles, and that

intercept the same arc are congruent

• AA similarity principle

• Definition of similar triangles

• Cross products

Secant segments theorem

Theorem: If two secants intersect in an exterior of a circle, then the product of the measures of one secant and its external secant segment is equal to the product of the measures of the other secant and its external secant segment so: AC*AB=AE*DE

Secant segments theorem

Theorem: AB*AC=AD*AE

First we form two triangles: ACD and AEB to help in the proof.

Paragraph proof:

AC and AE are secant segments. By the Reflective Property angles BAD and DAB are congruent.

Inscribed angles that intercept the same arcs are congruent. So ACD is congruent to AEB. By the definition of similar triangles, AB/AD=AE/AC.

Since the cross products of a proportion are similar , AB*AC=AD*AE.

Proportional Perimeter Theorem

• If two triangles are similar, then the perimeters are proportional to the measures of the corresponding sides

Creating illustrations in GeoGebra as aid in proofs

• Discuss how creating the illustrations can help to understand the problem and help in the proof