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Geometry: Similar Triangles

Geometry: Similar Triangles. MA.912.G.4.5 Apply theorems involving segments divided proportionally. Block 28. Similar triangles. Definition: Two triangles are similar if and only if their corresponding angles are congruent and corresponding sides are proportional.

emery-faulkner
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Geometry: Similar Triangles

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  1. Geometry: Similar Triangles

  2. MA.912.G.4.5 Apply theorems involving segments divided proportionally Block 28

  3. Similar triangles Definition: Two triangles are similar if and only if their corresponding angles are congruent and corresponding sides are proportional.

  4. 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.

  5. There is a lot of situations where similar triangles naturally arise

  6. 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

  7. Proofs using similar triangles

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

  9. Two-column proof: • Refer to the following picture:

  10. 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

  11. 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

  12. Secant segments theorem Theorem: AB*AC=AD*AE

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

  14. 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.

  15. Proportional Perimeter Theorem • If two triangles are similar, then the perimeters are proportional to the measures of the corresponding sides

  16. Creating illustrations in GeoGebra as aid in proofs • Discuss how creating the illustrations can help to understand the problem and help in the proof

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