3.2: Similarity

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3.2.1 Preparation Theorems

For △ABC construct line ℓ such that ℓ || AC and B is on ℓ. What is the relationship between the three angles at (smaller than a straight angle) to the angles of the triangle?

Theorem: Triangle Angle Sum

The angle sum of the interior angles of a triangle is π.

Corollary: Euclidean Exterior Angle

The measure of an exterior angle of a triangle is equal to the sum of the two opposing, interior angles.

Definition: Parallelogram

A quadrilateral is a parallelogram if and only if both opposing pairs of sides are parallel.

Theorem

Opposite sides of a parallelogram are congruent.

Theorem

If a transversal intersects three parallel lines in such a way as to divide itself into congruent segments, then any transversal of these parallel lines is also divided into congruent segments.

3.2.2 Explore Similarity Theorems

Definition: Altitude

A line segment is an altitude if it connects a vertex of a triangle to the foot of the perpendicular on the opposite side.

Construct a triangle and enough parallel lines to divide one side of the triangle into four equal parts. Into how many parts do these lines divide the other sides?

Definition: Triangle Area

The area of a triangle is equal to one half of the product of one side times the length of the altitude from the opposing vertex to that side.

Construct △ABC. m> Construct DE such that B-D-A, B-E-C and DE || AC.

Construct AE and EF such that F is the foot of the perpendicular from E. Reduce the ratio of the areas of △DEB and △AED.
Construct DC and DG such that G is the foot of the perpendicular from D. Reduce the ratio of the areas of △DEB and △CDE.
Prove area of △AED is equal to the area of △CDE.

3.2.3 Similarity Theorems

Definition: Similar Triangles

Two triangles are similar if and only if corresponding angles are congruent and the ratio of corresponding sides is constant.

Definition

A line parallel to one side of a triangle and intersecting the other two sides divides those sides proportionally.

Definition

If a line parallel to one side of a triangle divides the other sides proportionally then the two triangles (large and part) are similar.

Theorem: Angle-Angle-Angle (AAA) Similarity

Two triangle are similar if and only if they have three congruent angles.

3.2.4 Extending Similarity

Construct a definition for similar quadrilaterals. Construct examples to show that your definition works.

Explain why similarity is not defined simply as "all angles are congruent."