Similarity

Section 3.2 Similarity

Subsection 3.2.1 Preparation Theorems

Checkpoint 3.2.1.

For \(\triangle ABC\) construct line \(\ell\) such that \(\ell \parallel \overline{AC}\) and \(B\) is on \(\ell.\) What is the relationship between the three angles at \(B\) (smaller than a straight angle) to the angles of the triangle?

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Theorem 3.2.2. Triangle Angle Sum.

The angle sum of the interior angles of a triangle is \(\pi.\)

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Corollary 3.2.3. Euclidean Exterior Angle.

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

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Definition 3.2.4. Parallelogram.

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

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Theorem 3.2.5.

Opposite sides of a parallelogram are congruent.

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Theorem 3.2.6.

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.

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Subsection 3.2.2 Explore Similarity Theorems

Definition 3.2.7. 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.

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Checkpoint 3.2.8.

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?

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Definition 3.2.9. 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.

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Checkpoint 3.2.10.

Construct \(\triangle ABC\text{.}\) Construct \(\overline{DE}\) such that \(B-D-A,\) \(B-E-C\text{,}\) and \(\overline{DE} \parallel \overline{AC}.\)

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Construct \(\overline{AE}\) and \(\overline{EF}\) such that \(F\) is the foot of the perpendicular from \(E.\) Reduce the ratio of the areas of \(\triangle DEB\) and \(\triangle AED.\)

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Construct \(\overline{DC}\) and \(\overline{DG}\) such that \(G\) is the foot of the perpendicular from \(D.\) Reduce the ratio of the areas of \(\triangle DEB\) and \(\triangle CDE.\)

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Prove area of \(\triangle AED\) is equal to the area of \(\triangle CDE.\)

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Subsection 3.2.3 Similarity Theorems

Definition 3.2.11. Similar Triangles.

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

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Lemma 3.2.12.

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

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Corollary 3.2.13.

A line parallel to one side of a triangle divides the triangle into two (whole and part), similar triangles.

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Theorem 3.2.14. Angle-Angle-Angle (AAA) Similarity.

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

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Subsection 3.2.4 Extending Similarity

Checkpoint 3.2.15.

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

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Checkpoint 3.2.16.

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

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