Illustration showing four angles that can be used to define different relationships, such as adjacent, supplementary, etc..

Relationships Between 4 Angles

Illustration showing four angles that can be used to define different relationships, such as adjacent,…

Illustration showing six angles that can be used to define different relationships, such as adjacent, supplementary, etc..

Relationships Between 6 Angles

Illustration showing six angles that can be used to define different relationships, such as adjacent,…

An illustration showing when one straight line meets another straight line and makes the adjacent angles equal, each of these angles is called a right angle.

Right Angles With Same Vertex

An illustration showing when one straight line meets another straight line and makes the adjacent angles…

An illustration of angles with the same vertex made up of 4 lines.

Angles With Same Vertex

An illustration of angles with the same vertex made up of 4 lines.

Illustration showing that the sum of angle 1 and angle 2 is angle ABC.

Sum of Angles

Illustration showing that the sum of angle 1 and angle 2 is angle ABC.

Illustration of the superposition of two angles to compare magnitudes of the same kind.

Superposition of Angles

Illustration of the superposition of two angles to compare magnitudes of the same kind.

Illustration of supplementary angles. Two angles whose sum is a straight angle.

Supplementary Angles

Illustration of supplementary angles. Two angles whose sum is a straight angle.

Illustration of supplementary angles. Two angles whose sum is a straight angle. This can be used tow show the Theorem: If two adjacent angles have their exterior sides in a straight line, these angles are supplementary.

Supplementary Angles

Illustration of supplementary angles. Two angles whose sum is a straight angle. This can be used tow…

Illustration of supplementary angles. Two angles whose sum is a straight angle. This can be used tow show the Theorem: If two adjacent angles are supplementary, their exterior sides are in the same straight line.

Supplementary Angles

Illustration of supplementary angles. Two angles whose sum is a straight angle. This can be used tow…

Illustration of supplementary angles. Angles LMK and KMH are supplementary.

Supplementary Angles

Illustration of supplementary angles. Angles LMK and KMH are supplementary.

Illustration showing that angles 1 and 2 are supplementary.

Supplementary Angles

Illustration showing that angles 1 and 2 are supplementary.

Illustration showing angles 1 and 2 are supplementary and angles ACD and DCB are supplementary. Also, Angles ACD and DCB are right angles.

Supplementary and Right Angles

Illustration showing angles 1 and 2 are supplementary and angles ACD and DCB are supplementary. Also,…

Vertical angles are when two angles have the same vertex, and the sides of the one are prolongations of the sides of the other. In this illustration, a and b are vertical angles and c and d are vertical angles.

Vertical Angles

Vertical angles are when two angles have the same vertex, and the sides of the one are prolongations…

Illustration showing that angles 1 and 2 are vertical and angles 3 and 4 are vertical.

Vertical Angles

Illustration showing that angles 1 and 2 are vertical and angles 3 and 4 are vertical.

Anglesite.

Anglesite

Anglesite.

"Hexagonal; tri-pyramidal. Crystals usually long prismatic in habit; sometimes short prismatic or tabular. Usually terminated by prominent pyramid of first order and frequently a basal plane." — Ford, 1912

Apatite

"Hexagonal; tri-pyramidal. Crystals usually long prismatic in habit; sometimes short prismatic or tabular.…

"Hexagonal; tri-pyramidal. Crystals usually long prismatic in habit; sometimes short prismatic or tabular. Usually terminated by prominent pyramid of first order and frequently a basal plane." — Ford, 1912

Apatite

"Hexagonal; tri-pyramidal. Crystals usually long prismatic in habit; sometimes short prismatic or tabular.…

Apophyllite.

Apophyllite

Apophyllite.

Apophyllite.

Apophyllite

Apophyllite.

"Tetragonal. Usually shows a combination of prism of second order, pyramid of first and basal plane." — Ford, 1912

Apophyllite

"Tetragonal. Usually shows a combination of prism of second order, pyramid of first and basal plane."…

"Orthorhombic. Acicular pyramidal; consisting of a prism terminated by a combination of a very steep pyramid and brachydome." — Ford, 1912

Aragonite

"Orthorhombic. Acicular pyramidal; consisting of a prism terminated by a combination of a very steep…

"Orthorhombic. Tabular; consisting of prominent brachypinacoid faces modified by a prism and a low brachydome." — Ford, 1912

Aragonite

"Orthorhombic. Tabular; consisting of prominent brachypinacoid faces modified by a prism and a low brachydome."…

Illustration used to show how to bisect a given arc.

Bisecting an Arc

Illustration used to show how to bisect a given arc.

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arch.

Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arch.

Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arch.

Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arch.

Arch

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle, we can find the areas of as many triangles as we have made from our circle. Therefore, to find the area of a circle: Find the area of one of the triangles and multiply by the number of triangles." -Foster, 1921

Area of Circle with Triangles

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle,…

Illustration showing how to find the area of a hexagon using the triangles that make it up.

Area of Hexagon

Illustration showing how to find the area of a hexagon using the triangles that make it up.

"It is found that the cord covering the curved surface is twice as long as the one covering the flat surface. So the area of the entire curved surface of a sphere is equal to the area of the surface of 4 circles like the one measured." -Foster, 1921

Area of Sphere

"It is found that the cord covering the curved surface is twice as long as the one covering the flat…

"The bases of the pyramids are considered as forming the surface of the sphere, while the altitude of the pyramids is the radius of the sphere." -Foster, 1921

Area of Sphere

"The bases of the pyramids are considered as forming the surface of the sphere, while the altitude of…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arrow pointing left.

Arrow Pointing Left

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arrow pointing left.

Arrow Pointing Left

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arrow pointing left.

Arrow Pointing Left

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an arrow pointing left.

Arrow Pointing Left

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

"Orthohombric. Usually in tabular diamond-shaped crystals, formed by a short prism terminated by low brachydomes. The brachydomes are usually striated parallel to the brachy-axis." — Ford, 1912

Arsenopyrite

"Orthohombric. Usually in tabular diamond-shaped crystals, formed by a short prism terminated by low…

Axinite.

Axinite

Axinite.

"Triclinic. Crystals usually thin with sharp edges but varied in habit." — Ford, 1912

Axinite

"Triclinic. Crystals usually thin with sharp edges but varied in habit." — Ford, 1912

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry axis

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

Barite.

Barite

Barite.

"Orthohombic. Crystals usually tabular parallel to base; often diamond shaped because of the presence of a short prism. Both macro- and brachydomes usually present, either beveling the corners of the diamond-shaped crystals, or if the prism faces are wanting, beveling the edges of the tavles and forming rectangular prismatic-shaped crystals elongated parallel to either the brachy- or macro-axis." — Ford, 1912

Barite

"Orthohombic. Crystals usually tabular parallel to base; often diamond shaped because of the presence…

"Orthohombic. Crystals usually tabular parallel to base; often diamond shaped because of the presence of a short prism. Both macro- and brachydomes usually present, either beveling the corners of the diamond-shaped crystals, or if the prism faces are wanting, beveling the edges of the tavles and forming rectangular prismatic-shaped crystals elongated parallel to either the brachy- or macro-axis." — Ford, 1912

Barite

"Orthohombic. Crystals usually tabular parallel to base; often diamond shaped because of the presence…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bear.

Bear

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bear.

Bear

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bear.

Bear

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bear.

Bear

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bearded man in a tophat.

Bearded Man in a Tophat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bearded man in a tophat.

Bearded Man in a Tophat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bearded man in tophat.

Bearded Man in Tophat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a bearded man in tophat.

Bearded Man in Tophat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

"Hexagonal. Strong prismatic habit. Frequently vertically striated and grooved. Forms usually present consist only of prism of first order and base." — Ford, 1912

Beryl

"Hexagonal. Strong prismatic habit. Frequently vertically striated and grooved. Forms usually present…

"Hexagonal. Strong prismatic habit. Frequently vertically striated and grooved. Forms usually present consist only of prism of first order and base. Small pyramid faces of both the first and second orders sometimes occur, but the pyramid faces are rarely prominent." — Ford, 1912

Beryl

"Hexagonal. Strong prismatic habit. Frequently vertically striated and grooved. Forms usually present…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a beta fish.

Beta Fish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a beta fish.

Beta Fish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a beta fish.

Beta Fish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a beta fish.

Beta Fish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Illustration showing that the bisector of a given angle is the locus of points equidistant from the sides of the angle.

Angle Bisector

Illustration showing that the bisector of a given angle is the locus of points equidistant from the…

Illustration showing that the perpendicular bisector of a given line is the locus of points equidistant from the extremities of the line.

Perpendicular Bisector of a Line

Illustration showing that the perpendicular bisector of a given line is the locus of points equidistant…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a boat.

Boat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a boat.

Boat

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…