Glossary

7.3 Similarity proofs and applications

2 min read•july 22, 2024

Similar triangles are like twins with different sizes. They share the same shape but can have different measurements. This concept is super useful for solving real-world problems, like figuring out how tall a building is without climbing it.

Triangle similarity theorems are the secret sauce for proving triangles are similar. These rules help us compare triangles and find missing measurements. Understanding these theorems opens up a world of problem-solving possibilities in geometry.

Similarity Theorems and Proofs

Triangle similarity theorems

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Top images from around the web for Triangle similarity theorems

  • Geometry for Elementary School/The Side-Angle-Angle congruence theorem - Wikibooks, open books ... View original

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  • Non-right Triangles: Law of Sines – Algebra and Trigonometry OpenStax View original

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  • AA (Angle-Angle) Similarity Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar (△ABC∼△DEF\triangle ABC \sim \triangle DEF if ∠A≅∠D\angle A \cong \angle D and ∠B≅∠E\angle B \cong \angle E)
  • SAS (Side-Angle-Side) Similarity Theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar (△ABC∼△DEF\triangle ABC \sim \triangle DEF if ABDE=BCEF\frac{AB}{DE} = \frac{BC}{EF} and ∠B≅∠E\angle B \cong \angle E)
  • SSS (Side-Side-Side) Similarity Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar (△ABC∼△DEF\triangle ABC \sim \triangle DEF if ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF})
  • These theorems provide the foundation for proving the similarity of triangles and are essential for solving problems involving similar triangles (right triangles, isosceles triangles)

Applications of similarity proofs

  • Proportional sides in similar triangles means that corresponding sides of similar triangles are proportional (△ABC∼△DEF\triangle ABC \sim \triangle DEF, then ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF})
  • Proportional altitudes, medians, angle bisectors, and perimeters in similar triangles means that these segments and measurements are proportional to their corresponding sides
  • have congruent corresponding angles and proportional corresponding sides, allowing for the application of similarity proofs to solve problems involving various geometric shapes (rectangles, pentagons)

Similar figures in real-world problems

  • Indirect measurement uses similar triangles to find the height of tall objects (buildings, trees) or the distance across wide spaces (rivers, canyons) by setting up proportions using corresponding sides of similar triangles
  • and maps utilize the properties of similar figures to determine actual distances and sizes using scale factors and proportions (architectural models, topographic maps)
  • Shadow problems involve using similar triangles formed by objects and their shadows to find unknown heights or distances (flagpoles, monuments)

Similarity vs proportionality

  • Proportional lengths in similar figures means that corresponding lengths are proportional, with the (k) being the ratio of corresponding lengths
  • Proportional areas in similar figures means that the areas are proportional to the square of the scale factor (Area1Area2=k2\frac{Area_1}{Area_2} = k^2)
  • Proportional volumes in similar figures means that the volumes are proportional to the cube of the scale factor (Volume1Volume2=k3\frac{Volume_1}{Volume_2} = k^3)
  • Understanding the relationship between similarity and proportionality is crucial for solving problems involving scale factors, ratios, and dimensional analysis (enlargements, reductions)

Key Terms to Review (19)

AA Similarity: AA similarity, or Angle-Angle similarity, states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. This means that their corresponding sides are in proportion and their corresponding angles are equal. This concept is fundamental in establishing similarity between triangles and is essential in solving problems involving similar figures and trigonometric ratios.
Altitude Theorem: The Altitude Theorem states that the length of the altitude drawn from a vertex of a triangle to the opposite side creates two right triangles that are similar to the original triangle and to each other. This theorem connects the concepts of altitudes, similarity, and triangle properties, allowing us to establish relationships between the sides and angles of triangles when an altitude is present.
Angle Bisector Theorem: The Angle Bisector Theorem states that if a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. This theorem not only highlights the relationship between angles and distances but also plays a significant role in various geometric proofs and properties, such as triangle inequalities, relationships in circles, and establishing similarity in figures.
Congruence of Angles: Congruence of angles refers to the property where two angles have the same measure, often denoted using the symbol '≅'. This concept is fundamental in geometry as it establishes the relationship between angles in various figures and is crucial for proving similarity and congruence in triangles and other shapes.
Congruence of Corresponding Angles: The congruence of corresponding angles refers to the principle that when two parallel lines are intersected by a transversal, the angles that occupy the same relative position at each intersection are equal. This concept is crucial in establishing relationships between angles in geometric proofs and applications, especially regarding similarity and congruence of triangles.
Corresponding sides ratio: The corresponding sides ratio refers to the relationship between the lengths of corresponding sides of similar figures, which are proportional to one another. This ratio is fundamental in establishing similarity between geometric shapes, as it helps to show that two figures have the same shape but may differ in size. By comparing the lengths of corresponding sides, we can determine scale factors and apply them in real-world situations such as model building or map reading.
Dilations: Dilations are transformations that produce an image that is the same shape as the original figure but is a different size. This transformation involves enlarging or reducing a figure by a scale factor relative to a fixed point called the center of dilation, and it preserves the angles of the original figure while changing the lengths of the sides proportionally.
Flowchart Proof: A flowchart proof is a visual representation of logical reasoning used to demonstrate the validity of a mathematical statement or theorem. This method organizes steps, reasons, and conclusions in a clear, sequential format, making it easier to follow the logical flow of an argument. Flowchart proofs are particularly useful in geometry for illustrating similarity proofs and applications, as they help clarify relationships between figures.
Map reading: Map reading is the skill of interpreting and understanding the information presented on a map, including the analysis of scales, symbols, and geographic features. This skill is crucial for navigating space and understanding spatial relationships, which directly connects to concepts like similarity in geometry where proportions and comparative measurements are essential.
Proportional lengths of sides: Proportional lengths of sides refer to the relationship between the lengths of corresponding sides of similar figures, where these lengths maintain a constant ratio. This concept is crucial in establishing similarity between geometric shapes, allowing for comparisons of size without altering the shape. When two figures are similar, their corresponding side lengths are proportional, meaning that the ratio of one pair of sides is equal to the ratio of another pair of sides.
Proportional Parts: Proportional parts refer to segments, lengths, or other measurable attributes that maintain a constant ratio to each other, especially in the context of similar figures. In geometry, when two triangles are similar, the lengths of their corresponding sides are proportional, meaning that the ratios of the lengths are equal. This concept is essential for understanding similarity proofs and applications, as it helps establish relationships between figures and calculate unknown measures based on known ones.
Sas similarity: SAS similarity, or Side-Angle-Side similarity, is a concept in geometry that states two triangles are similar if two sides of one triangle are proportional to two sides of another triangle, and the included angles between those sides are equal. This principle is essential for establishing the similarity of triangles, which allows for the use of proportions to find unknown lengths or angles. By understanding SAS similarity, you can apply it to various scenarios involving right triangles and solve problems that require an understanding of trigonometric ratios.
Scale Factor: A scale factor is a number that describes how much a figure is enlarged or reduced in size. It is essential in understanding the relationship between similar figures, as it helps determine proportional dimensions in transformations, allowing us to compare their sizes and shapes effectively.
Scale Models: Scale models are physical representations of objects that are proportionally reduced or enlarged to a certain ratio compared to the actual object. They maintain the same proportions as the original, allowing for analysis and understanding of complex structures in a more manageable form. Scale models play a significant role in similarity proofs and applications, as they help visualize and solve geometric problems involving similar shapes and ratios.
Similar Polygons: Similar polygons are geometric figures that have the same shape but not necessarily the same size. This means that corresponding angles in similar polygons are equal, and the lengths of corresponding sides are in proportion. Understanding similar polygons is essential for solving problems involving proportions, ratios, and applications in real-world contexts, such as architecture and design.
SSS Similarity: SSS Similarity, or Side-Side-Side Similarity, states that if the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. This concept is crucial for proving triangle similarity through geometric relationships, allowing us to infer corresponding angle equality based on side length ratios.
Transitive Property of Similarity: The transitive property of similarity states that if two figures are similar to a third figure, then they are similar to each other. This property is fundamental in proving relationships between geometric figures, allowing for the establishment of similarity through indirect comparisons. Understanding this concept is essential for solving problems involving proportions, ratios, and real-life applications of similar figures.
Triangle Proportionality Theorem: The Triangle Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. This theorem establishes a crucial relationship between ratios in triangles, connecting ideas of similarity, dilations, and proportional relationships in geometric figures. Understanding this theorem is vital for solving problems involving similar triangles, proving congruency, and applying concepts in right triangles.
Two-column proof: A two-column proof is a structured way of presenting a mathematical argument where statements and corresponding reasons are organized into two columns. This format helps clarify the logical flow of the proof and ensures that each step is justified by a reason, making it an essential tool for demonstrating the validity of geometric concepts and relationships.
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