Illustrative Math Alignment: Grade 7 Unit 7

In this program we explore the properties of angles and planes. We do this in the context of two real-world applications. In the first, we explore Japan's Himeji Castle and in the process learn about different types of angles and how they're used in a defensive fortification. In the second applicaiton we explore sedimentary rock layers as examples of parallel planes. We explore the Burgess Shale fossils.

The observatory in Arecibo, Puerto Rico provides astronomers insights into the structure of our solar system. Geometrically, the solar system relies on the plane known as the ecliptic. In studying the Earth's orbit it is important to know that the Earth’s axis of rotation is at an angle relative to the ecliptic. This segment introduces the key themes of the program.

Himeji castle in Japan is a marvel of architecture and a startling example of geometry and military science. The castle was used to protect samurai armies from invading forces, and the use of acute, obtuse, and right angles as part of the defense structure provide many opportunities for exploring the nature of geometric angles.

In the Canadian Rockies, the Burgess Shale fossils provide a window to prehistoric Earth. Fossil layers are folded into sedimentary rocks. And sedimentary rocks are examples of parallel planes. This segment uses the properties of planes to analyze fossils.

In this program we explore the properties of triangle. We do this in the context of two real-world applications. In the first, we explore the triangular trusses in the Eiffel Tower and in the process learn about key properties of triangles. In the second application, we look at right-triangle-shaped sails on sail boat and why these are the ideal shape for efficient sailing.

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.

The Eiffel Tower includes quite a number of exposed triangular trusses. The properties of triangles are used to explore and explain the frequent use of triangular trusses in many building. In particular, isosceles and equilateral triangular trusses are explored. In addition triangle postulates and similarity are explored and analyzed.

The ancient port city of Corinth in Greece allows us to explore the evolution of sailing. The shapes of sails went from rectangular to triangular. The use of right-triangle-shaped sails changed the nature of sailing. Becoming less reliant on oarsmen and being able to sail under nearly any wind conditions, the sailboat made the conquest of the seas possible. We explore the properties of right triangles that make such sails ideal.

Video Tutorial: Ratios and Proportions: Scale Drawings. In this video, we look at the properties of similar figures and use proportions to find lengths of corresponding sides.

Topic

Angles

Definition

An acute angle is an angle that measures less than 90 degrees.

Topic

Angles

Definition

Adjacent angles are two angles that share a common vertex and side, but do not overlap.

Topic

Angles

Definition

Alternate exterior angles are pairs of angles formed when a transversal crosses two parallel lines, and they lie on opposite sides of the transversal and outside the parallel lines.

Topic

Angles

Definition

Alternate interior angles are pairs of angles formed when a transversal crosses two parallel lines, and they lie on opposite sides of the transversal and inside the parallel lines.

Topic

Angles

Definition

An angle is formed by two rays with a common endpoint, known as the vertex.

Description

Angles are a fundamental concept in geometry, forming the basis for understanding shapes and their properties. They are used in a wide range of applications, from measuring land and constructing buildings to designing machinery and creating art. Angles are measured in degrees or radians, and their understanding is essential for solving problems involving triangles, circles, and other geometric figures. In education, angles are introduced early in the curriculum to develop spatial awareness and geometric reasoning, providing a foundation for more advanced mathematical concepts.

Topic

Angles

Definition

An angle bisector is a line or ray that divides an angle into two equal parts.

Description

Angle bisectors are used in various geometric constructions and proofs, such as constructing the incenter of a triangle, which is the point where the angle bisectors of a triangle intersect. In practical applications, angle bisectors are used in design and architecture to create symmetrical and balanced structures. For students, learning about angle bisectors enhances their understanding of geometric constructions and the properties of angles, contributing to their overall mathematical proficiency.

Topic

Angles

Definition

Angle measure refers to the size of an angle, typically expressed in degrees or radians.

Description

Understanding angle measurement is crucial for solving geometric problems and for applications in trigonometry, physics, and engineering. Accurate angle measurement is essential in fields such as surveying, navigation, and construction, where precise calculations are necessary for successful outcomes. In education, mastering angle measurement is a key component of the geometry curriculum, helping students develop skills in spatial reasoning and problem-solving.

Topic

Angles

Definition

The vertex of an angle is the common endpoint of the two rays that form the angle.

Topic

Angles

Definition

A central angle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle.

Description

Central angles are important in the study of circles and are used to define arcs and sectors. They are commonly used in problems involving circular motion and in the design of circular objects, such as wheels and gears. In mathematics education, understanding central angles helps students grasp the properties of circles and their applications, forming a basis for more advanced studies in trigonometry and calculus.

Topic

Angles

Definition

A circumscribed angle is an angle formed outside a circle by two intersecting lines, each tangent to the circle.

Description

Circumscribed angles are used in geometry to study the properties of tangents and their relationships with circles. They are important in problems involving circle theorems and are applied in fields such as architecture and design, where circular shapes are common. For students, learning about circumscribed angles enhances their understanding of circle geometry and the relationships between angles and arcs, contributing to their overall mathematical knowledge.

Topic

Angles

Definition

Complementary angles are two angles whose measures add up to 90 degrees.

Topic

Angles

Definition

Congruent angles are angles that have the same measure.

Description

Congruent angles are a key concept in geometry, often used in proving the similarity and congruence of geometric figures. They are essential in understanding the properties of polygons, where congruent angles indicate symmetry and balance. In real-world applications, congruent angles are used in design and engineering to ensure that components are identical and fit together seamlessly. For students, learning about congruent angles enhances their understanding of geometric relationships and proofs, contributing to their overall mathematical proficiency.

Topic

Angles

Definition

Corresponding angles are pairs of angles that are in similar positions at each intersection where a transversal crosses two lines.

Topic

Angles

Definition

An exterior angle is an angle formed outside a polygon by extending one of its sides.

Topic

Angles

Definition

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint.

Description

Inscribed angles are important in the study of circles, where they are used to define arcs and sectors. They are commonly used in problems involving circle theorems and in the design of circular objects, such as wheels and gears. In mathematics education, understanding inscribed angles helps students grasp the properties of circles and their applications, forming a basis for more advanced studies in trigonometry and calculus.

Topic

Angles

Definition

An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees.

Topic

Angles

Definition

Opposite interior angles are pairs of angles opposite the exterior angle of a triangle. In the case of parallel lines cut by a transversal, they are congruent. 

Topic

Angles

Definition

A reflex angle is an angle that measures more than 180 degrees but less than 360 degrees.

Description

Reflex angles are often encountered in the study of rotations and circular motion, where they are used to describe angles that extend beyond a straight line. They are important in fields such as physics and engineering, where understanding rotations and angular measurements is crucial. In mathematics education, reflex angles help students develop a comprehensive understanding of angle measurement and classification, contributing to their overall geometric reasoning and problem-solving abilities.

Topic

Angles

Definition

A right angle is an angle that measures exactly 90 degrees.

Description

Right angles are fundamental in geometry and are often used as a reference for measuring other angles. They are commonly found in various geometric shapes and real-world structures, such as squares, rectangles, and buildings. In real-world applications, right angles are essential in construction and design, where they ensure that elements are perpendicular and aligned correctly. In education, understanding right angles is crucial for learning about perpendicularity and the properties of geometric figures, forming a basis for more advanced mathematical concepts.

Topic

Angles

Definition

A straight angle is an angle that measures exactly 180 degrees.

Description

Straight angles are used to describe angles that form a straight line, often encountered in the study of linear relationships and transformations. They are important in fields such as physics and engineering, where understanding linear motion and alignment is crucial. In mathematics education, straight angles help students develop a comprehensive understanding of angle measurement and classification, contributing to their overall geometric reasoning and problem-solving abilities.

Topic

Angles

Definition

Supplementary angles are two angles whose measures add up to 180 degrees.

Topic

Angles

Definition

Vertical angles are pairs of opposite angles formed by two intersecting lines, and they are equal in measure.

Topic

Geometry Basics

Definition

Corresponding angles are angles that are in the same relative position at each intersection where a straight line crosses two others.

Description

Corresponding angles are formed when a transversal intersects two parallel lines. These angles are equal in measure, which is a key concept in proving the properties of parallel lines and solving geometric problems. For example, if two parallel lines are cut by a transversal, then each pair of corresponding angles is equal. This concept is used extensively in geometry to establish relationships between angles and to solve complex problems involving parallel lines.

Topic

Polygons

Definition

An exterior angle of a polygon is the angle formed between any side of the polygon and the extension of an adjacent side.

Topic

Polygons

Definition

The sum of interior angles of a polygon with n sides is given by the formula: (n - 2) × 180°, where n is the number of sides in the polygon.

Description

The sum of interior angles is a fundamental property of polygons that plays a crucial role in geometry. This concept is essential for understanding the structure and properties of various polygons, from simple triangles to complex multi-sided figures. The formula (n - 2) × 180° provides a quick and efficient way to calculate the total measure of all interior angles in any polygon, regardless of its regularity.

Topic

Polygons

Definition

The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360°.

Description

The sum of exterior angles of a polygon is a fundamental concept in geometry that holds true for all polygons, regardless of their shape or number of sides. This property is crucial for understanding the nature of polygons and their relationship to circular motion. Exterior angles are formed by extending each side of the polygon and measuring the angle between this extension and the adjacent side.

Topic

Polygons

Definition

An interior angle of a polygon is an angle formed inside the polygon by two adjacent sides.

Topic

Quadrilaterals

Definition

The sum of the interior angles of a quadrilateral is always 360 degrees.

This is part of a collection of definitions of geometric theorems and postulates.

This is part of a collection of definitions of geometric theorems and postulates.

This is part of a collection of definitions of geometric theorems and postulates.

This is part of a collection of definitions of geometric theorems and postulates.

This is part of a collection of definitions of geometric theorems and postulates.

The following section includes background information on angles. 

What Are Angles?

In this illustration two rays intersect at a point. Points A and B are on one ray and points B and C are on another ray. These two rays form an angle.

This is part of a collection of definitions of geometric theorems and postulates.

Topic

Triangles

Definition

A 30-60-90 right triangle is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees.

Description

The 30-60-90 triangle is a special right triangle characterized by its angle measures. The sides of this triangle are in the ratio 1:√3:2, meaning the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg. This relationship can be derived using trigonometric principles and is fundamental in geometry.

Topic

Triangles

Definition

A 45-45-90 right triangle is a right triangle with two 45-degree angles and one 90-degree angle.

Description

The 45-45-90 triangle is another special right triangle where the two legs are of equal length, and the hypotenuse is √2 times the length of each leg. This type of triangle is also known as an isosceles right triangle due to its equal legs.

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.

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.

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.

This is the Teacher's Guide that accompanies Geometry Applications: Angles and Planes.

Related Resources

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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.