Illustrative Math Alignment: Grade 8 Unit 2

Dilations, Similarity, and Introducing Slope

Lesson 7: Similar Polygons

Use the following Media4Math resources with this Illustrative Math lesson.

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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10 Topic Geometric Shapes Description Determine if two right triangles are similar. The triangle on the left has angle 40°, and the other triangle has a 60° angle. For similarity, all corresponding angles must be congruent. The angles in each triangle do not match, so the triangles aren't similar. Therefore, the answer is No, the triangles aren't similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10, 8, and 12, while triangle DEF has sides 5, 3, and 6. Triangles are similar if all corresponding sides are proportional. Only two pairs of sides here have a 2:1 ratio, so the triangles are not similar. Therefore, the answer is No, the triangles arenÕt similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Both triangles have angles 60°, 57°, and 63°. Triangles are similar if corresponding angles are equal. Since all corresponding angles match, the triangles are similar. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has angles 75°, 42°, and an unknown angle, and triangle DEF has angles 75° and 63° and an unknown angles. Triangles are similar if all corresponding angles are congruent. Solving for the unknown angles confirms angle congruency, so the triangles are similar. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5 Topic Geometric Shapes Description Determine if two right triangles, ABC and DEF, are similar. Triangle ABC has angles 90° and 40°, and triangle DEF has angles 90° and 50°. Triangles are similar if all corresponding angles are congruent. Solving reveals congruent corresponding angles. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has a right angle and an angle of 40°, and triangle DEF has a right angle and 45°. Not all corresponding angles are congruent, so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 45° angle. Triangles are similar if two pairs of sides are proportional, and the included angle is congruent. The side ratio is 2:1, and both included angles are 45°. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 40° angle. Triangles are similar if two pairs of sides are proportional, and the included angles are congruent. Here, the included angles differ (45° vs. 40°), so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9 Topic Geometric Shapes Description Determine if two right triangles are similar. The triangle on the left has sides 10 and 15 and an included right angle, while the other triangle has sides 5 and 7.5 and an included right angle. Triangles are similar if two pairs of corresponding sides are proportional, and corresponding included angles are congruent. The side ratio is 2:1, and the included right angles are congruent. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Polygons--Polygon Classification: Example 1	Math Example--Polygons--Polygon Classification: Example 1 Topic Polygons Description This example showcases a pentagon with all sides labeled as 5. It demonstrates a regular pentagon, characterized by five sides of equal length. This visual representation helps students understand the concept of regular polygons and their defining characteristics. The topic of polygon classification is crucial in geometry. This collection of examples provides a comprehensive look at various types of pentagons, both regular and irregular. By examining different configurations of sides and angles, students can develop a deeper understanding of polygon properties and classification criteria.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 10	Math Example--Polygons--Polygon Classification: Example 10 Topic Polygons Description This example showcases a pentagon labeled with vertices A, B, C, D, E, where the angles are marked as 110°, 95°, 120°, 107°, and 108°. It demonstrates an irregular pentagon based on the variation in angle measures, highlighting that regularity depends on both side lengths and angle measures. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples emphasizes the importance of both side lengths and angle measures in determining the regularity of pentagons.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 11	Math Example--Polygons--Polygon Classification: Example 11 Topic Polygons Description This example presents a pentagon labeled with vertices A, B, C, D, E, where the sides are marked with variables w, x, y, z, and v. It illustrates an irregular pentagon by using different variables to represent potentially different side lengths, emphasizing that side length variation results in irregularity. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides a comprehensive look at various types of pentagons, highlighting the importance of side length equality in determining regularity.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 12	Math Example--Polygons--Polygon Classification: Example 12 Topic Polygons Description This example features a pentagon labeled with vertices A, B, C, D, E, where the angles are marked as 108°, x, y, z, and 107°. It demonstrates an irregular pentagon based on the variation in angle measures, highlighting that regularity depends on all angles being equal. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples emphasizes the importance of angle measures in determining the regularity of pentagons, showing that even one different angle makes the polygon irregular.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 13	Math Example--Polygons--Polygon Classification: Example 13 Topic Polygons Description This example showcases a hexagon with all sides labeled as 5, indicating equal length. It demonstrates a regular hexagon, characterized by six sides of equal length. This visual representation helps students understand the concept of regular polygons and their defining characteristics. The topic of polygon classification is crucial in geometry. This collection of examples provides a comprehensive look at various types of hexagons, both regular and irregular. By examining different configurations of sides and angles, students can develop a deeper understanding of polygon properties and classification criteria.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 14	Math Example--Polygons--Polygon Classification: Example 14 Topic Polygons Description This example presents a hexagon with all sides labeled as x, indicating equal length. It illustrates another instance of a regular hexagon, reinforcing the concept that a regular polygon has all sides of equal length, even when represented by a variable. Polygon classification is a fundamental topic in geometry that helps students understand the properties and characteristics of different shapes. This collection of examples provides a comprehensive overview of various hexagon types, allowing students to distinguish between regular and irregular polygons based on their side lengths and angle measures.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 15	Math Example--Polygons--Polygon Classification: Example 15 Topic Polygons Description This example features a hexagon with sides labeled y, x, and 5, showing that x = y = 5, making all sides equal in length. It demonstrates a regular hexagon by using a combination of variables and numeric values to represent equal side lengths. Understanding polygon classification is crucial in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights different ways to represent regular hexagons, emphasizing the importance of side length equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 16	Math Example--Polygons--Polygon Classification: Example 16 Topic Polygons Description This example showcases a hexagon with all interior angles labeled as 120 degrees, indicating equal angles. It demonstrates that a regular hexagon can be identified not only by equal side lengths but also by equal angle measures, reinforcing the dual criteria for regularity in polygons. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides a comprehensive look at various aspects of regular hexagons, emphasizing both side length and angle measure as defining characteristics.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 17	Math Example--Polygons--Polygon Classification: Example 17 Topic Polygons Description This example presents a hexagon with six equal sides labeled as x and equal angles, indicating a regular hexagon. It reinforces the concept of a regular polygon by visually representing both the equality of all sides and angles. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples provides various representations of regular hexagons, emphasizing the importance of both side length and angle measure equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 18	Math Example--Polygons--Polygon Classification: Example 18 Topic Polygons Description This example features a hexagon with two angles labeled as 120 degrees and sides labeled x and y, where x = y. It demonstrates how to determine if a hexagon is regular by solving for the unknown angle measures and confirming equal side lengths. Polygon classification is a crucial topic in geometry that helps students distinguish between regular and irregular shapes. This collection of examples provides a comprehensive look at various types of hexagons, highlighting the importance of both angle measures and side lengths in determining regularity.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 19	Math Example--Polygons--Polygon Classification: Example 19 Topic Polygons Description This example showcases a hexagon with sides labeled with varying lengths: AB = 6, BC = 4, CD = 6, DE = 3, EF = 6, and FA = 5. It demonstrates an irregular hexagon, emphasizing that side length variation results in irregularity. Understanding polygon classification is crucial in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights the differences between regular and irregular hexagons, focusing on side length as a key determining factor.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 2	Math Example--Polygons--Polygon Classification: Example 2 Topic Polygons Description This example presents a pentagon with all sides labeled as x, indicating equal length. It illustrates another instance of a regular pentagon, reinforcing the concept that a regular polygon has all sides of equal length. Polygon classification is a fundamental topic in geometry that helps students understand the properties and characteristics of different shapes. This collection of examples provides a comprehensive overview of various pentagon types, allowing students to distinguish between regular and irregular polygons based on their side lengths and angle measures.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 20	Math Example--Polygons--Polygon Classification: Example 20 Topic Polygons Description This example presents a hexagon with sides labeled as x, y, and z, indicating varying lengths. It illustrates another case of an irregular hexagon, demonstrating that the presence of different variables for side lengths suggests irregularity. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides a comprehensive look at various types of hexagons, emphasizing the importance of side length equality in determining regularity.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 21	Math Example--Polygons--Polygon Classification: Example 21 This is part of a collection of math examples that focus on geometric shapes.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 22	Math Example--Polygons--Polygon Classification: Example 22 This is part of a collection of math examples that focus on geometric shapes.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 23	Math Example--Polygons--Polygon Classification: Example 23 This is part of a collection of math examples that focus on geometric shapes.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 24	Math Example--Polygons--Polygon Classification: Example 24 This is part of a collection of math examples that focus on geometric shapes.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 25	Math Example--Polygons--Polygon Classification: Example 25 Topic Polygons Description This example showcases an octagon with all sides labeled as 5, indicating equal length. It demonstrates a regular octagon, characterized by eight sides of equal length. This visual representation helps students understand the concept of regular polygons and their defining characteristics. The topic of polygon classification is crucial in geometry. This collection of examples provides a comprehensive look at various types of octagons, both regular and irregular. By examining different configurations of sides and angles, students can develop a deeper understanding of polygon properties and classification criteria.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 26	Math Example--Polygons--Polygon Classification: Example 26 Topic Polygons Description This example presents an octagon with all sides labeled as x, indicating equal length. It illustrates another instance of a regular octagon, reinforcing the concept that a regular polygon has all sides of equal length, even when represented by a variable. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples provides various representations of regular octagons, emphasizing the importance of side length equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 27	Math Example--Polygons--Polygon Classification: Example 27 Topic Polygons Description This example features an octagon with sides labeled as x, z, and 5, showing that x = z = 5, indicating equal length for all sides. It demonstrates a regular octagon by using a combination of variables and numeric values to represent equal side lengths. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides a comprehensive look at various aspects of regular octagons, emphasizing the importance of side length equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 28	Math Example--Polygons--Polygon Classification: Example 28 Topic Polygons Description This example showcases an octagon with all angles labeled as 135 degrees, indicating congruent angles. It demonstrates that a regular octagon can be identified not only by equal side lengths but also by equal angle measures, reinforcing the dual criteria for regularity in polygons. Understanding polygon classification is crucial in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights different aspects of regular octagons, emphasizing both side length and angle measure as defining characteristics.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 29	Math Example--Polygons--Polygon Classification: Example 29 Topic Polygons Description This example presents an octagon with all angles labeled as "x," indicating a regular polygon. It illustrates a regular octagon by visually representing the equality of all angles using a single variable. This reinforces the concept that a regular polygon has all angles of equal measure. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides a comprehensive look at various aspects of regular octagons, emphasizing the importance of side length equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 3	Math Example--Polygons--Polygon Classification: Example 3 Topic Polygons Description This example features a pentagon with sides labeled x and two sides labeled as 5, with x = 5, demonstrating that all sides are equal. It reinforces the concept of a regular pentagon by showing how different notations can represent the same length. The study of polygon classification helps students develop critical thinking skills in geometry. This collection of examples illustrates various ways to represent and identify regular pentagons, emphasizing the importance of side length equality in classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 30	Math Example--Polygons--Polygon Classification: Example 30 Topic Polygons Description This example features an octagon with angles labeled as 135° and angles labeled as "u" and "x." It demonstrates how to determine if an octagon is regular by solving for the unknown angle measures and confirming equal side lengths. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples provides insights into how partial information about angle measures and side lengths can be used to determine the regularity of octagons.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 31	Math Example--Polygons--Polygon Classification: Example 31 Topic Polygons Description This example showcases an irregular octagon with sides of varying lengths labeled as 4, 5, 6, and 3. It demonstrates that not all eight-sided figures are regular, emphasizing the importance of equal side lengths in the classification of regular polygons. Polygon classification is a crucial topic in geometry that helps students distinguish between regular and irregular shapes. This collection of examples provides a comprehensive look at various types of octagons, highlighting the differences between regular and irregular polygons based on side lengths.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 32	Math Example--Polygons--Polygon Classification: Example 32 Topic Polygons Description This example presents an irregular octagon with sides labeled as w, x, y, z, and u, indicating varying lengths. It illustrates another case of an irregular octagon, demonstrating that the presence of different variables for side lengths suggests irregularity. Understanding polygon classification is crucial in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights the importance of considering side lengths when determining the regularity of octagons.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 33	Math Example--Polygons--Polygon Classification: Example 33 Topic Polygons Description This example features an octagon with sides labeled as x, y, 5, and 3. The sides are of varying lengths, indicating it is irregular. It demonstrates how a combination of variables and specific numeric values can be used to show irregularity in polygons. Polygon classification is a fundamental concept in geometry that helps students analyze and categorize shapes based on their properties. This collection of examples provides insights into how partial information about side lengths can be used to determine the irregularity of octagons.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 34	Math Example--Polygons--Polygon Classification: Example 34 Topic Polygons Description This example showcases an octagon with angles labeled as 150°, 140°, 115°, 145°, 155°, 115°, 125°, and 135°. The angles vary, indicating it is irregular. It demonstrates that irregularity in polygons can be determined by examining angle measures alone. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights the importance of angle measures in determining the regularity of octagons, showing that even if side lengths are not given, angle information can be sufficient for classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 35	Math Example--Polygons--Polygon Classification: Example 35 Topic Polygons Description This example presents an octagon with sides labeled as x, y, z, w, u, t, s, and v. The sides are of varying lengths and angles, indicating it is irregular. It illustrates how using different variables for both side lengths and angles can suggest irregularity in a polygon. Polygon classification is a crucial topic in geometry that helps students distinguish between regular and irregular shapes. This collection of examples provides a comprehensive look at various types of octagons, highlighting the importance of considering both side lengths and angle measures in determining regularity.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 36	Math Example--Polygons--Polygon Classification: Example 36 Topic Polygons Description This example features an octagon with angles labeled as 155° and 125°, and other angles labeled with variables t, v, w, x, y, and z. The angles vary, indicating it is irregular. It demonstrates how a combination of specific angle measures and variables can be used to show irregularity in polygons. Understanding polygon classification is essential in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights the importance of angle measures in determining the regularity of octagons, showing that even partial information can be sufficient for classification.	Definition of a Polygon
Math Example--Polygons--Polygon Classification: Example 4	Math Example--Polygons--Polygon Classification: Example 4 Topic Polygons Description This example showcases a pentagon with each angle labeled as 108 degrees, indicating congruent angles. It demonstrates that a regular pentagon can be identified not only by equal side lengths but also by equal angle measures. Understanding polygon classification is crucial in geometry as it helps students recognize and categorize shapes based on their properties. This collection of examples highlights different aspects of regular pentagons, emphasizing both side length and angle measure as defining characteristics.	Definition of a Polygon

Illustrative Math Alignment: Grade 8 Unit 2

Dilations, Similarity, and Introducing Slope

Lesson 7: Similar Polygons

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