Straight line AB is perpendicular to the lines in plane MN at point B. "If from the foot of a perpendicular to a plane a straight line is drawn at right angles to any line in the plane, the line drawn from its intersection with the line in the plane to any point of the perpendicular is perpendicular to the line of the plane."

Line Perpendicular to Plane

Straight line AB is perpendicular to the lines in plane MN at point B. "If from the foot of a perpendicular…

Two straight lines, AB and CD, perpendicular to the same plane (MN) are parallel.

Lines Perpendicular to Plane

Two straight lines, AB and CD, perpendicular to the same plane (MN) are parallel.

Diagram used to prove the theorem: "A plane perpendicular to a radius at its extremity is tangent to the sphere."

Sphere Tangent to Plane

Diagram used to prove the theorem: "A plane perpendicular to a radius at its extremity is tangent to…

Illustration of three intersecting planes. "The intersections of two parallel planes by a third plane are parallel lines."

3 Intersecting Planes

Illustration of three intersecting planes. "The intersections of two parallel planes by a third plane…

Illustration of three intersecting planes. "The straight line perpendicular to one of two parallel lanes is perpendicular to the other."

3 Intersecting Planes

Illustration of three intersecting planes. "The straight line perpendicular to one of two parallel lanes…

"If two angles not in the same plane have their sides respectively parallel and lying in the same direction, they are equal and their planes are parallel."

Angles In Parallel Planes

"If two angles not in the same plane have their sides respectively parallel and lying in the same direction,…

Two intersecting planes on a third plane. "If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane."

Intersecting Planes

Two intersecting planes on a third plane. "If two intersecting planes are each perpendicular to a third…

"If two intersecting lines are each parallel to a given plane, the plane of these lines is parallel to the given plane."

Parallel Planes

"If two intersecting lines are each parallel to a given plane, the plane of these lines is parallel…

CD is the projection of AB upon OX. In each case AE=CD and AE=ABcos theta.

Projection of a Line Segment

CD is the projection of AB upon OX. In each case AE=CD and AE=ABcos theta.

Illustration used to prove "Two angles whose sides are perpendicular each to each are either equal or supplementary."

Angles With Perpendicular Sides Are Equal or Supplementary Proof

Illustration used to prove "Two angles whose sides are perpendicular each to each are either equal or…

Illustration of a rectangle, which is a parallelogram with all angles right angles.

Rectangle

Illustration of a rectangle, which is a parallelogram with all angles right angles.

"Let a represent a, b water, and c a piece of glass. The ray d, striking each medium in a perpendicular direction." -Comstock 1850

Refraction, air, water, glass

"Let a represent a, b water, and c a piece of glass. The ray d, striking each medium in a perpendicular…

"Let the medium b be glass, and the medium c, water. The ray a, as it falls upon the medium b, is refracted towards the perpendicular line e d; but when it enters the water, whose refractive power is less than that of glass, it is not bent so near the perpendicular as before, and hence it is refracted from, instead of towards the perpendicular line, and approaches the originial direction of the ray a g, when passing through the air." -Comstock 1850

Refraction, glass and water

"Let the medium b be glass, and the medium c, water. The ray a, as it falls upon the medium b, is refracted…

A heraldic shield with a green surface, which is represented by the perpendicular lines, drawn from the head to the base of the shield.

Gules Shield

A heraldic shield with a green surface, which is represented by the perpendicular lines, drawn from…

A heraldic shield with a black surface, which is shown by the crossing horizontal and vertical lines.

Sable Shield

A heraldic shield with a black surface, which is shown by the crossing horizontal and vertical lines.

The lines of contact are equal, but the axes are neither parallel to each other, or the line of contact.

Lateral Sliding Contact

The lines of contact are equal, but the axes are neither parallel to each other, or the line of contact.

Illustration of a square, a rectangle with all sides equal.

Square

Illustration of a square, a rectangle with all sides equal.

Diagram used to prove the theorem: "Between two straight lines not in the same plane only one perpendicular can be drawn."

Perpendicular Between 2 Straight Lines

Diagram used to prove the theorem: "Between two straight lines not in the same plane only one perpendicular…

Supplementary Angles degrees 90 and 90

Supplementary Angles 90 - 90

Supplementary Angles degrees 90 and 90

Illustration of radius drawn to point of contact of a tangent.

Point of Tangency

Illustration of radius drawn to point of contact of a tangent.

Draftsman's method to draw a tangent, AB, to a circle

Tangent To A Circle

Draftsman's method to draw a tangent, AB, to a circle

Draftsman's method to draw a line tangent to two given circles

Tangent To Two Circles

Draftsman's method to draw a line tangent to two given circles

Illustration of a leaning tower with a perpendicular drawn from the top of the tower to the ground to form a right triangle.

Leaning Tower

Illustration of a leaning tower with a perpendicular drawn from the top of the tower to the ground to…

Illustration of a palm tree that is perpendicular to the ground. The tree is perfectly straight, as is the ground. This drawing could be used for shadow, proportion, trigonometric, or Pythagorean Theorem problems.

Palm Tree Perpendicular to Ground

Illustration of a palm tree that is perpendicular to the ground. The tree is perfectly straight, as…

Triangle ABC with sides a,b,c labeled and angles A,B,C labeled and h labeled as height.

Triangle ABC With Height h

Triangle ABC with sides a,b,c labeled and angles A,B,C labeled and h labeled as height.

A test to ensure the triangle is accurate by drawing a perpendicular line from a T-square.

Triangle Accuracy Test using T-square

A test to ensure the triangle is accurate by drawing a perpendicular line from a T-square.

Illustration used to show "The altitudes of a triangle are concurrent."

Altitudes In A Triangle

Illustration used to show "The altitudes of a triangle are concurrent."

Illustration used to prove "The bisectors of the angles of a triangle are concurrent in a point which is equidistant from the three sides of the triangle."

Angle Bisectors In A Triangle

Illustration used to prove "The bisectors of the angles of a triangle are concurrent in a point which…

Illustration to show the bisector of the vertical angle of a triangle makes with the perpendicular from the vertex to the base an angle equal to half the difference of the base angles.

Triangle With Bisector and Perpendicular From Vertex Drawn

Illustration to show the bisector of the vertical angle of a triangle makes with the perpendicular from…

Triangle ABC with sides a,b,c labeled and angles A,B,C labeled and h labeled as height.

Obtuse Triangle ABC With Height h

Triangle ABC with sides a,b,c labeled and angles A,B,C labeled and h labeled as height.

Illustration to show that the perpendicular bisector of the base of an isosceles triangle passes through the vertex and bisects the angle at the vertex.

Perpendicular Bisector of the Base of an Isosceles Triangle

Illustration to show that the perpendicular bisector of the base of an isosceles triangle passes through…

Illustration used to prove "The perpendicular bisectors of the sides of a triangle are concurrent in a point which is equidistant from the three vertices of the triangle."

Perpendicular Bisectors In A Triangle

Illustration used to prove "The perpendicular bisectors of the sides of a triangle are concurrent in…

Illustration to show the perpendiculars dropped from any point in the equilateral triangle to the three sides is constant, and equal to the altitude.

Perpendiculars Dropped Any Point in the Equilateral Triangle

Illustration to show the perpendiculars dropped from any point in the equilateral triangle to the three…

Illustration to show the perpendiculars dropped from the midpoint of the base to the legs of an isosceles triangle are equal.

Perpendiculars Dropped From Midpoint of Base to Legs of Isosceles Triangles

Illustration to show the perpendiculars dropped from the midpoint of the base to the legs of an isosceles…

Illustration showing a right triangle (one that has one right angle).

Right Triangle

Illustration showing a right triangle (one that has one right angle).

Right triangle OCB that can be used to show the relationships between x, y, r, and Θ.

Right Triangle OCB With, x, y, and r shown

Right triangle OCB that can be used to show the relationships between x, y, r, and Θ.