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In this section, we learn that triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Key theorems related to the similarity of triangles are introduced, including the Basic Proportionality Theorem, as well as criteria such as AAA, AA, SSS, and SAS for proving triangle similarity.
Two triangles are deemed similar under certain conditions concerning their angles and sides. To be similar, two triangles must: 1. Have all their corresponding angles equal. 2. Have their corresponding sides in proportion.
If two triangles meet these criteria, they are considered similar, denoted as ΞABC ~ ΞDEF, indicating that triangle ABC is similar to triangle DEF.
This section sets the groundwork for understanding triangle similarity, which is essential for applying geometric principles in real-world scenarios, such as those encountered in trigonometry and architectural design.
Triangle Similarity: Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
Angle-Angle (AA) Criterion: Two triangles are similar if two angles of one triangle are equal to two angles of another triangle.
Side-Side-Side (SSS) Criterion: Two triangles are similar if the lengths of their corresponding sides are proportional.
Side-Angle-Side (SAS) Criterion: Two triangles are similar if one angle is equal to an angle in the other triangle and the sides including these angles are proportional.
Basic Proportionality Theorem: A line drawn parallel to one side of a triangle splits the other two sides proportionately.
Triangles with angles the same, their shape will stay the same game.
Imagine two trees growing at the same angle on a hill, both having their shadows painted by the evening sunβthis shows how similarity works, just like triangle proportions.
Remember 'S.A.S' for sides and angle, if they match up, similarity's a good tangle!
Using triangle similarity to find heights of objects by measuring shadows can help in real-life applications like surveying.
If angle A in triangle ABC is 60Β° and angle D in triangle DEF is also 60Β°, and the sides AB and DE are in ratio 2:3, then triangles ABC and DEF are similar by the SAS criterion.
Term: Similar Figures
Definition: Figures that have the same shape but not necessarily the same size.
Figures that have the same shape but not necessarily the same size.
Term: Thales' Theorem
Definition: A theorem stating that if two triangles are equiangular, the ratios of their corresponding sides are equal.
A theorem stating that if two triangles are equiangular, the ratios of their corresponding sides are equal.
Term: Proportionality
Definition: A relationship between two quantities such that they maintain a constant ratio.
A relationship between two quantities such that they maintain a constant ratio.
Term: Basic Proportionality Theorem
Definition: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Term: AAA criterion
Definition: A criterion for similarity stating that if all three angles of one triangle are equal to the corresponding angles of another triangle, then the triangles are similar.
A criterion for similarity stating that if all three angles of one triangle are equal to the corresponding angles of another triangle, then the triangles are similar.
Term: SSS criterion
Definition: A criterion for similarity stating that if the sides of one triangle are in proportion to the sides of another triangle, then the triangles are similar.
A criterion for similarity stating that if the sides of one triangle are in proportion to the sides of another triangle, then the triangles are similar.
Term: SAS criterion
Definition: A criterion for similarity stating that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.
A criterion for similarity stating that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.