Illustrative Math Alignment: Grade 7 Unit 7

Angles, Triangles, and Prisms

Lesson 8: Triangles with 3 Common Measures

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Nodes
Definition--Triangle Concepts--Centroid of a Triangle	Centroid of a Triangle Topic Triangles Definition The centroid of a triangle is the point where its three medians intersect, often considered the triangle's center of mass. Description The centroid of a triangle is a significant point in geometry, representing the intersection of the triangle's medians. Each median is a line segment drawn from a vertex to the midpoint of the opposite side. The centroid is also known as the triangle's center of mass or balance point.	Definition of a Triangle
Definition--Triangle Concepts--Circle Inscribed In a Triangle	Circle Inscribed In a Triangle Topic Triangles Definition A circle inscribed in a triangle is a circle that touches all three sides of the triangle from the inside. Description An inscribed circle in a triangle, also known as an incircle, is a circle that is tangent to all three sides of the triangle. The center of this circle is called the incenter, which is the point where the angle bisectors of the triangle intersect.	Definition of a Triangle
Definition--Triangle Concepts--Circumcenter of a Triangle	Circumcenter of a Triangle Topic Triangles Definition The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect, and it is the center of the triangle's circumcircle. Description The circumcenter of a triangle is a crucial concept in geometry, representing the point where the perpendicular bisectors of the triangle's sides intersect. This point is equidistant from all three vertices of the triangle, making it the center of the circumcircle, which is the circle that passes through all three vertices.	Definition of a Triangle
Definition--Triangle Concepts--Congruent Triangles	Congruent Triangles Topic Triangles Definition Congruent triangles are triangles that have the same size and shape, with corresponding sides and angles that are equal. Description Congruent triangles are fundamental in geometry, representing triangles that are identical in size and shape. This means that all corresponding sides and angles of congruent triangles are equal. The concept of congruence is crucial for proving various geometric theorems and properties.	Definition of a Triangle
Definition--Triangle Concepts--Equilateral Triangle	Equilateral Triangle Topic Triangles Definition An equilateral triangle is a triangle with all three sides of equal length and all three angles equal to 60 degrees. Description An equilateral triangle is a special type of triangle where all three sides are of equal length, and all three interior angles are equal to 60 degrees. This symmetry makes equilateral triangles unique and significant in various geometric constructions.	Definition of a Triangle
Definition--Triangle Concepts--Hypotenuse	Hypotenuse Topic Triangles Definition The hypotenuse is the longest side of a right triangle, opposite the right angle. Description The hypotenuse is a crucial concept in geometry, representing the longest side of a right triangle, opposite the right angle. The Pythagorean theorem relates the hypotenuse to the other two sides of the triangle: a2 + b2 = c2 where c is the hypotenuse.	Definition of a Triangle
Definition--Triangle Concepts--Incenter of a Triangle	Incenter of a Triangle Topic Triangles Definition The incenter of a triangle is the point where the angle bisectors of the triangle intersect, and it is the center of the triangle's incircle. Description The incenter of a triangle is a significant point in geometry, representing the intersection of the triangle's angle bisectors. This point is equidistant from all three sides of the triangle, making it the center of the incircle, which is the circle that is tangent to all three sides.	Definition of a Triangle
Definition--Triangle Concepts--Inscribed Triangle	Inscribed Triangle Topic Triangles Definition An inscribed triangle is a triangle that is drawn inside a circle, with all its vertices touching the circle. Description An inscribed triangle, also known as a circumtriangle, is a triangle that is drawn inside a circle, with all its vertices touching the circle. The circle is called the circumcircle, and its center is the circumcenter of the triangle.	Definition of a Triangle
Definition--Triangle Concepts--Interior Angles of a Triangle	Interior Angles of a Triangle Topic Triangles Definition The interior angles of a triangle are the angles inside the triangle, and their sum is always 180 degrees. Description The interior angles of a triangle are a fundamental concept in geometry, representing the angles inside the triangle. The sum of the interior angles of any triangle is always 180 degrees, which is a crucial property used in various geometric proofs and constructions.	Definition of a Triangle
Definition--Triangle Concepts--Isosceles Triangle, Definition 1	Isosceles Triangle, Definition 1 Topic Triangles Definition An isosceles triangle is a triangle with at least two sides of equal length. Description An isosceles triangle is a type of triangle that has at least two sides of equal length. This property gives the isosceles triangle its unique characteristics, such as having two equal angles opposite the equal sides. This concept is significant in various real-world applications, such as in design and architecture, where symmetry and balance are important. For example, understanding the properties of isosceles triangles helps in creating aesthetically pleasing and structurally sound designs.	Definition of a Triangle
Definition--Triangle Concepts--Isosceles Triangle, Definition 2	Isosceles Triangle, Definition 2 Topic Triangles Definition An isosceles triangle is a triangle with two sides of equal length and two equal angles opposite those sides. Description An isosceles triangle is a type of triangle that has two sides of equal length and two equal angles opposite those sides. This symmetry gives the isosceles triangle its unique properties and makes it significant in various geometric constructions.	Definition of a Triangle
Definition--Triangle Concepts--Legs of a Right Triangle	Legs of a Right Triangle Topic Triangles Definition The legs of a right triangle are the two sides that form the right angle. Description The legs of a right triangle are the two sides that form the right angle. These sides are essential in the application of the Pythagorean theorem, which relates the lengths of the legs to the length of the hypotenuse: a2 + b2 = c2	Definition of a Triangle
Definition--Triangle Concepts--Medians of a Triangle	Medians of a Triangle Topic Triangles Definition The medians of a triangle are line segments drawn from each vertex to the midpoint of the opposite side. Description The medians of a triangle are significant in geometry, representing line segments drawn from each vertex to the midpoint of the opposite side. The point where the medians intersect is called the centroid, which is the triangle's center of mass.	Definition of a Triangle
Definition--Triangle Concepts--Obtuse Triangle	Obtuse Triangle Topic Triangles Definition An obtuse triangle is a triangle with one angle greater than 90 degrees. Description An obtuse triangle is characterized by having one of its interior angles greater than 90 degrees. This type of triangle is significant in geometry as it represents a unique class of triangles with specific properties. In real-world applications, obtuse triangles are used in various design and architectural projects where specific angle measurements are required. For example, understanding the properties of obtuse triangles helps in creating structures with unique shapes and angles.	Definition of a Triangle
Definition--Triangle Concepts--Orthocenter of a Triangle	Orthocenter of a Triangle Topic Triangles Definition The orthocenter of a triangle is the point where the three altitudes intersect. Description The orthocenter of a triangle is a significant point in geometry, representing the intersection of the triangle's three altitudes. Each altitude is a perpendicular line segment drawn from a vertex to the opposite side. This concept is essential in various real-world applications, such as in design and engineering, where precise measurements and symmetry are crucial. For example, understanding the orthocenter helps in creating balanced and structurally sound designs.	Definition of a Triangle
Definition--Triangle Concepts--Perimeter of a Triangle	Perimeter of a Triangle Topic Triangles Definition The perimeter of a triangle is the sum of the lengths of its three sides. Description The perimeter of a triangle is a fundamental concept in geometry, representing the total distance around the triangle. The perimeter is calculated by adding the lengths of the triangle's three sides. This concept is essential in various real-world applications, such as in land measurement, construction, and design. For example, calculating the perimeter of a triangular plot of land helps in land valuation and planning. In construction, understanding the perimeter is crucial for determining the amount of materials needed for a project.	Definition of a Triangle
Definition--Triangle Concepts--Pythagorean Triples	Pythagorean Triples Topic Triangles Definition Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem: a2 + b2 = c2 Description Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Common examples include (3, 4, 5) and (5, 12, 13).	Definition of a Triangle
Definition--Triangle Concepts--Right Triangle	Right Triangle Topic Triangles Definition A right triangle is a triangle with one 90-degree angle. Description A right triangle is a type of triangle that has one angle measuring 90 degrees. This type of triangle is fundamental in geometry and trigonometry, as it forms the basis for the Pythagorean theorem: a2 + b2 = c2 This concept is essential in various real-world applications, such as in construction and navigation. For example, right triangles are used to create right angles and ensure accuracy in building projects. They also play a crucial role in various mathematical problems and proofs.	Definition of a Triangle
Definition--Triangle Concepts--Scalene Triangle	Scalene Triangle Topic Triangles Definition A scalene triangle is a triangle with all three sides of different lengths and all three angles of different measures. Description A scalene triangle is characterized by having all three sides of different lengths and all three angles of different measures. This type of triangle is significant in geometry as it represents a unique class of triangles with specific properties.	Definition of a Triangle
Definition--Triangle Concepts--Similar Triangles	Similar Triangles Topic Triangles Definition Similar triangles are triangles that have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional. Description Similar triangles are fundamental in geometry, representing triangles that have the same shape but not necessarily the same size. This means that all corresponding angles are equal, and all corresponding sides are proportional.	Definition of a Triangle
Definition--Triangle Concepts--Triangle Altitude	Triangle Altitude Topic Triangles Definition The altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Description The altitude of a triangle is a significant concept in geometry, representing the perpendicular distance from a vertex to the line containing the opposite side. This measurement is crucial in various geometric calculations, such as finding the area of a triangle.	Definition of a Triangle
Definition--Triangle Concepts--Triangle Height	Triangle Height Topic Triangles Definition The height of a triangle is the perpendicular distance from a vertex to the line containing the opposite side, often used interchangeably with altitude. Description The height of a triangle, often used interchangeably with altitude, is a crucial concept in geometry. It represents the perpendicular distance from a vertex to the line containing the opposite side. This measurement is essential for calculating the area of a triangle, using the formula: Area= 1/2 × base × height	Definition of a Triangle
Definition--Triangle Concepts--Triangle Inscribed in a Circle	Triangle Inscribed in a Circle Topic Triangles Definition A triangle inscribed in a circle is a triangle whose vertices all lie on the circumference of the circle. Description A triangle inscribed in a circle, also known as a circumtriangle, is a triangle whose vertices all lie on the circumference of the circle. The circle is called the circumcircle, and its center is the circumcenter of the triangle.	Definition of a Triangle
Definition--Triangle Concepts--Triangle, Definition 1	Triangle, Definition 1 Topic Triangles Definition A triangle is a polygon with three edges and three vertices. Description A triangle is one of the simplest and most fundamental shapes in geometry, characterized by having three sides and three vertices. The sum of the interior angles of a triangle is always 180 degrees, which is a key property used in various geometric proofs and constructions. In real-world applications, triangles are used extensively in engineering, architecture, and design due to their inherent stability and strength. For example, triangular structures are often used in bridges and roof trusses to distribute weight evenly and provide support.	Definition of a Triangle
Definition--Triangle Concepts--Triangle, Definition 2	Triangle, Definition 2 Topic Triangles Definition A triangle is a three-sided polygon with three vertices and three angles. Description A triangle is a three-sided polygon that is fundamental in the study of geometry. Each triangle has three vertices and three angles, and the sum of its interior angles is always 180 degrees. This property is crucial for understanding various geometric principles and solving related problems.	Definition of a Triangle
Definition--Triangle Concepts--Vertex of a Triangle	Vertex of a Triangle Topic Triangles Definition A vertex of a triangle is a point where two sides of the triangle meet. Description A vertex of a triangle is one of the three points where the sides of the triangle intersect. Each triangle has three vertices, and these points are crucial for defining the shape and properties of the triangle.	Definition of a Triangle
Definition--Vector Concepts--Subtracting Vectors Using the Triangle Method	Definition--Vector Concepts--Subtracting Vectors Using the Triangle Method This is part of a collection of definitions on the topic of Vectors. —Click on PREVIEW to see the definition card.—	Vectors
Formulas--Area of a Right Triangle	Area of a Right Triangle This is the formula for finding the area of a right triangle. This is part of a collection of resources that include key math formulas. —CLICK ON THE PREVIEW BUTTON TO SEE THE IMAGE— To see the complete collection of formulas, click on this link. The following section provides background information on right triangles and area. Brief Review of Right Triangles A right triangle has certain properties regarding its angles and sides. A right triangle has a right angle and two acute angles. The right angle gives the right triangle a distinctive square corner.	Area and Perimeter of Triangles
Formulas--Area of a Triangle	Formulas--Area of a Triangle The formula for: Area of a Triangle. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Area and Perimeter of Triangles
Formulas--Area of an Equilateral Triangle	Formulas--Area of an Equilateral Triangle The formula for: Area of an Equilateral Triangle. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Area and Perimeter of Triangles
Formulas--Perimeter of a Triangle	Formulas--Perimeter of a Triangle The formula for the Perimeter of a Triangle. This is part of a collection of math formulas. To see the complete collection of formulas, click on this link. Note: The download is a JPG file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Area and Perimeter of Quadrilaterals
Geometry Applications, Triangles, Segment 1, Introduction	VIDEO: Geometry Applications: Triangles, Segment 1: Introduction The Bank of China building in Hong Kong is a dramatic example of triangular support. The notion of triangular trusses is introduced, along with the key concepts developed in the rest of the program.	Applications of Triangles and Definition of a Triangle
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Constructing an Isosceles Triangle	In this Slide Show, we show how to construct an isosceles triangle. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Definition of a Triangle and Geometric Constructions with Triangles
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: The Perpendicular Bisectors of a Triangle	In this Slide Show, we show how to construct the perpendicular bisectors of the three sides of a triangle to show they intersect at a single point. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Definition of a Triangle and Geometric Constructions with Triangles
INSTRUCTIONAL RESOURCE: TI-Nspire App Activity: Constructing an Isosceles Triangle	In this Slide Show, learn how to use the TI-Nspire App to construct an isosceles triangle. Note: The download is a PPT file.	Definition of a Triangle and Geometric Constructions with Triangles
Interactive Crossword Puzzle--Triangles	Interactive Crossword Puzzle--Triangles This interactive crossword puzzle tests knowledge of key terms on the topic of triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Math Game--Memory Game: Triangles	Interactive Math Game--Memory Game: Triangles Use this math game to review triangles. This is a Memory-style game in which students must remember the location of pairs of identical images. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Math Game: Math Riddles--Triangles	Interactive Math Game: Math Riddles--Triangles In this Math Riddles Game, have your students review vocabulary around the topic of triangles. The Math Riddles games are useful for practicing: Math Vocabulary, Key Concepts, Critical Thinking. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Word Search Puzzle--Triangles, Puzzle 1	Interactive Word Search Puzzle--Triangles, Puzzle 1 Solve an interactive word search puzzle on the topic of Triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Word Search Puzzle--Triangles, Puzzle 2	Interactive Word Search Puzzle--Triangles, Puzzle 2 Solve an interactive word search puzzle on the topic of Triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Word Search Puzzle--Triangles, Puzzle 3	Interactive Word Search Puzzle--Triangles, Puzzle 3 Solve an interactive word search puzzle on the topic of Triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Word Search Puzzle--Triangles, Puzzle 4	Interactive Word Search Puzzle--Triangles, Puzzle 4 Solve an interactive word search puzzle on the topic of Triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Interactive Word Search Puzzle--Triangles, Puzzle 5	Interactive Word Search Puzzle--Triangles, Puzzle 5 Solve an interactive word search puzzle on the topic of Triangles. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Definition of a Triangle
Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Labeled	Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Labeled This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.	Angles and Applications of Angles and Planes
Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Unlabeled	Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Unlabeled This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.	Angles and Applications of Angles and Planes
Math Clip Art--Geometry Basics--Classifying Triangles by Angles 01	Math Clip Art--Geometry Basics--Classifying Triangles by Angles 01 Topic Geometry Basics Description This math clip art image serves as the title card for a series on classifying triangles by angles. It reads "Classifying Triangles by Angle." This image is part of a comprehensive set designed to teach students about the different types of triangles based on their angles.	Definition of a Triangle
Math Clip Art--Geometry Basics--Classifying Triangles by Angles 02	Math Clip Art--Geometry Basics--Classifying Triangles by Angles 02 Topic Geometry Basics Description This math clip art image is the second in a series about categorizing angles in triangles. It depicts a triangle labeled ABC, illustrating the fundamental principle that a triangle includes three angles whose sum is 180 degrees. This key concept in geometry is visually represented, providing students with a clear understanding of this crucial property of triangles.	Definition of a Triangle
Math Clip Art--Geometry Basics--Classifying Triangles by Angles 03	Math Clip Art--Geometry Basics--Classifying Triangles by Angles 03 Topic Geometry Basics Description This math clip art image is the third in a series about categorizing angles in triangles. It showcases an acute triangle, demonstrating that an acute triangle consists of three acute angles. The image clearly illustrates the defining characteristic of an acute triangle, where all three angles are less than 90 degrees.	Definition of a Triangle
Math Clip Art--Geometry Basics--Classifying Triangles by Angles 04	Math Clip Art--Geometry Basics--Classifying Triangles by Angles 04 Topic Geometry Basics Description This math clip art image is the fourth in a series about categorizing angles in triangles. It presents a right triangle, illustrating that a right triangle consists of one 90-degree angle and two acute angles. The image clearly demonstrates the defining characteristic of a right triangle, emphasizing the 90-degree angle and the two acute angles that complete the 180-degree sum.	Definition of a Triangle
Math Clip Art--Geometry Basics--Classifying Triangles by Angles 05	Math Clip Art--Geometry Basics--Classifying Triangles by Angles 05 Topic Geometry Basics Description This math clip art image is the fifth in a series about categorizing angles in triangles. It showcases an obtuse triangle, demonstrating that an obtuse triangle consists of one obtuse angle and two acute angles. The image clearly illustrates the defining characteristic of an obtuse triangle, highlighting the one angle that is greater than 90 degrees and the two acute angles that complete the 180-degree sum.	Definition of a Triangle