CHAPTER 11 THE GEOMETRY OF THREE DIMENSIONS Section

CHAPTER 11 THE GEOMETRY OF THREE DIMENSIONS Section 11 -1 Points, Lines, and Planes Section 11 -2 Perpendicular Lines and Planes Section 11 -3 Parallel Lines and Planes C. N. Colón St. Barnabas H S Geometry

DO NOW Find the area of the rhombus: 10 10 12 12

Objectives: • You will review, develop and apply postulates for points lines and planes • Explore the properties of parallel and perpendicular lines and planes

RECALL THAT A POINT IS AN EXACT LOCATION IN SPACE. You are here.

A TRUE POINT HAS NO LENGTH NO WIDTH NO HEIGHT. In fact, you cannot see a true point.

A POINT IS NAMED BY A LETTER. Point P P

A line is a 1 -dimensional object that has only length r i d h t o b n s n o i ect i r e orev e n i l A f s e u n i t con

Because a line has no width or height, you cannot see a true line.

A LINE IS DEFINED BY 2 POINTS. A Line AB B

A RAY HAS A STARTING POINT BUT NO ENDING POINT.

JUST AS A RAY OF LIGHT HAS A STARTING POINT AND CONTINUES FOREVER IN THE SAME DIRECTION. SO TOO DOES A RAY IN GEOMETRY

OF COURSE, IF THE RAY HITS AN OBJECT, THE LIGHT COULD BE ABSORBED OR REFLECTED. but that is a topic for another course…. such as physics

A RAY IS ALSO DEFINED BY 2 POINTS. C Ray CD D

A LINE SEGMENT HAS A STARTING POINT AND AN ENDING POINT. LINE SEGMENTS CAN BE MEASURED.

A LINE SEGMENT IS ALSO DEFINED BY 2 POINTS. E Line Segment EF F

Ray BA A B To read a ray, start at the end point and go towards the arrow Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in opposite directions. A l B Line l or AB

LINES CAN DO VARIOUS THINGS.

LINES CAN INTERSECT AT A POINT TO FORM ANGLES. X Y Z

ANGLES ARE DEFINED BY 3 POINTS. X Y XYZ Z

Lines can intersect to form right angles 90

…AND THESE ANGLES ARE RIGHT ANGLES. R RST = 90⁰ USR = 90⁰ c S U T VST = 90⁰ USV = 90⁰ V

WHEN LINES INTERSECT TO FORM RIGHT ANGLES, THEY ARE SAID TO BE PERPENDICULAR. R c U UT RV S V T

LINES CAN INTERSECT TO FORM ACUTE AND OBTUSE ANGLES. 45 acute 135 obtuse

LINES CAN ALSO RUN INTO EACH OTHER TO FORM STRAIGHT ANGLES.

180 is a straight angle

LINES DO NOT ALWAYS INTERSECT.

LINES CAN BE PARALLEL.

Two non-vertical lines are parallel iff they have the same slope.

The Transitive Property of Parallel Lines Let ℓ 1, ℓ 2, and ℓ 3 be three coplanar lines. Prove that if ℓ 1 is parallel to ℓ 2 and ℓ 2 is parallel to ℓ 3 , then ℓ 1 is parallel to ℓ 3.

Given: ℓ 1║ℓ 2 and ℓ 2║ℓ 3 Prove: ℓ 1║ℓ 3 STATEMENT 1. ℓ 1║ℓ 2 and ℓ 2║ℓ 3 2. Let m 1, m 2, and m 3 be the slopes of ℓ 1, ℓ 2, and ℓ 3 3. m 1 = m 2 and m 2 = m 3 4. m 1 = m 3 5. ℓ 1 and ℓ 3 are parallel REASON 1. Given 2. Slopes of non-vertical lines are represented by m 3. Parallel lines have the same slope 4. Transitive Property of Equality 5. Parallel Lines have the same slope

ARE LINES THAT DO NOT INTERSECT ALWAYS PARALLEL? Believe it or not, it is possible for lines that do not intersect to also not be parallel.

LET’S CONSIDER A 3 -DIMENSIONAL OBJECT SUCH AS A CEREAL BOX. THIS SOLID HAS A SPECIAL NAME: RECTANGULAR PRISM.

Notice the 2 lines that do not intersect Notice that they also are not parallel.

Definition: Skew lines are lines in space that are neither parallel nor intersecting.

Two lines are SKEW iff they are not parallel and do not intersect. (For lines to be skew they must be in different planes

AT A BARBECUE A “SKEWER” MAY BE USED. S K E W E R

A SKEWER RAISES THE FOOD OFF THE SURFACE OF THE GRILL. IT IS NOT PARALLEL TO THE GRILL AND IT DOES NOT INTERSECT OR TOUCH THE GRILL.

IN A SOCCER FIELD THE GOAL POSTS AND SIDE LINES ARE SKEW LINES

HERE IS A RECTANGULAR PRISM WITH SKEW LINES DRAWN

SUMMARY OF LINES intersecting perpendicular 90º skew parallel

Identify the relationships between the lines that are formed by the edges of the cube There are many relationships: A pair of parallel lines: ____ and ____ A pair of perpendicular lines: ____ and ____ A pair of skew lines are: ____ and _____

A plane is a flat surface that has length & width but no height.

YOU CAN SEE A PLANE ONLY IF YOU VIEW IT AT A CERTAIN ANGLE.

A TRUE PLANE GOES ON FOREVER IN ALL DIRECTIONS.

A TRUE PLANE GOES ON FOREVER IN ALL DIRECTIONS.

A TRUE PLANE GOES ON FOREVER IN ALL DIRECTIONS.

A TRUE PLANE GOES ON FOREVER IN ALL DIRECTIONS.

A TRUE PLANE GOES ON FOREVER IN ALL DIRECTIONS.

A plane extends in two dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges. A B M C Plane M or plane ABC The 3 points are noncollinear

Theorem 11. 1 There is exactly one plane containing a line and a point not on the line A B C

Theorem 11. 2: Two intersecting lines determine a plane. Another way of saying this is if two lines intersect then there is exactly one plane containing them

The definition of parallel lines gives us another set of points that must lie in a plane. Definition: Parallel lines are lines that lie in the same plane and have no points in common. Thus, two lines are parallel iff they are coplanar and have no points in common.

The geometry of three dimensions is called solid geometry. Recall the coordinate plane where you had an x-axis and a y-axis.

The geometry of three dimensions involves three axes

If two planes intersect then they intersect in exactly one line

When planes intersect they form an angle.

Definition p. 424 A dihedral angle is the union of two half planes with a common edge.

Where you are sitting is made up of two half planes. These form an angle where you place yourself. The two half planes of a seat intersect at exactly one line.

AN X-WING FIGHTER FROM STAR WARS HAS WINGS THAT INTERSECT.

Planes can be perpendicular. Definition p. 424: Perpendicular planes are two planes that intersect to form a right dihedral angle

Planes can be parallel.

A TIE FIGHTER FROM STAR WARS HAS WINGS THAT ARE PARALLEL.

Parallel planes found in a rock formation on a beach south of Anchorage, Alaska.

Intersecting planes and parallel planes in an architectural design by Hawkins and Associates in Reno, Nevada

Intersecting planes in interior design The Atlas Chair by Scott Jarvie

Intersecting planes used in the description of the human body

“Two Intersecting Planes” by Sascha Ledinsky based on an M. C. Escher print.

INTERSECTING PLANES IN ORIGAMI

INTERSECTING PLANES IN FELT SPONGE ARTWORK

Intersecting planes of red blood cells.

Intersecting Planes in a molecule of water

Planes can be parallel intersecting perpendicular

Identify two planes that appear to have the given relationship. 1. parallel planes _______and _______ 2. perpendicular planes _______and _______ 3. neither parallel nor perpendicular _______ and ____

SKETCHING INTERSECTIONS A line that intersects a plane in one point Ø Ø Ø Draw a plane and a line. Emphasize the point where they meet. Dashes below the point indicate where the line is hidden by the plane

SKETCHING INTERSECTING PLANES

AND NOW --- A SUMMARY PLEASE do not memorize them - you must understand each of them! Go slowly and visualize each situation before answering any questions on a quiz, or on a test.

SUMMARY OF THEOREMS: CH. 11 -1 • There is one and only one plane containing three non-collinear points • A plane containing any two points contains all the points on the line determined by those two points • There is exactly one plane containing a line and a point not on the line • If two lines intersect, there is exactly one plane containing them, in other words, two intersecting lines determine a plane • DEFINITION: Parallel lines in space are lines in the same plane that have no points in common • DEFINITION: Skew lines are lines in space that are neither parallel nor intersecting

SUMMARY OF THEOREMS: CH 11 -2 • If two planes intersect, then they intersect in exactly one line. • DEFINTION: A dihedral angle is the union of two halfplanes with a common edge (line) • DEFINTION: The measure of a dihedral angle is the measure of the plane angle formed by two rays each in a different half-plane of the angle and each perpendicular to the common edge at the same point of the edge. • DEFINITION: Perpendicular planes are two planes that intersect to form a right dihedral angle • If a line not in a plane intersects the plane, then it intersects it in exactly one point. • DEFINITION: A line is perpendicular to a plane iff it is perpendicular to each line in the plane through the intersection of the line and the plane

SUMMARY OF THEOREMS: CH 11 -2 CONT’D • • • DEFINITION: A plane is perpendicular to a line if a line is perpendicular to a plane. At a given point on a line , there are infinitely many lines perpendicular to the given line If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by these lines. If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane. Conversely, if a plane contains a line perpendicular to another plane, then the planes are perpendicular DEFINITION: Two planes are perpendicular iff one plane contains a line perpendicular to the other. (biconditional)

SUMMARY OF THEOREMS: CH 11 -2 CONT’D • • Through a given point on a plane, there is only one line perpendicular to the given plane. Through a given point on a line, there can be only one plane perpendicular to the given line. If a line is perpendicular to a plane , then any line perpendicular to the given line at its point of intersection with the given plane, is in the plane. If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the plane.

SUMMARY OF THEOREMS: CH 11 -3 • Parallel planes are planes that have no point in common • If a plane intersects two parallel planes, then the intersection is two parallel lines • Two lines perpendicular to the same plane are parallel • Two lines perpendicular to the same plane are coplanar. • If two planes are perpendicular to the same line, then they are parallel – Conversely, if two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other. • Two planes are perpendicular to the same line iff the planes are parallel. (Biconditional) • DEFINITION: the distance between two planes is the length of the line segment perpendicular to both planes with an endpoint on each plane. • Parallel planes are everywhere equidistant

HOMEWORK p. 422 #4 -12 (e) p. 432 #4 -20 (mo 4) p. 439 # 4 -12 (mo 4)