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## The Geometry of Three Dimensions

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**Geometry Chap11**The Geometry of Three Dimensions Eleanor Roosevelt High School Chin-Sung Lin**ERHS Math Geometry**The Geometry of Three Dimensions • The geometry of three dimensions is called • solid geometry Mr. Chin-Sung Lin**ERHS Math Geometry**Points, Lines, and Planes Mr. Chin-Sung Lin**ERHS Math Geometry**Postulates of the Solid Geometry • There is one and only one plane containing three non-collinear points B A C Mr. Chin-Sung Lin**ERHS Math Geometry**Postulates of the Solid Geometry • A plane containing any two points contains all of the points on the line determined by those two points B A m Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of the Points, Lines & Planes • There is exactly one plane containing a line and a point not on the line B m A P Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of the Points, Lines & Planes • If two lines intersect, then there is exactly one plane containing them • Two intersecting lines determine a plane m P A B n Mr. Chin-Sung Lin**ERHS Math Geometry**Parallel Lines in Space • Lines in the same plane that have no points in common • Two lines are parallel if and only if they are coplanar and have no points in common m n Mr. Chin-Sung Lin**ERHS Math Geometry**Skew Lines in Space • Skew lines are lines in space that are neither parallel nor intersecting n m Mr. Chin-Sung Lin**ERHS Math Geometry**Example • Both intersecting lines and parallel lines lie in a plane • Skew lines do not lie in a plane • Identify the parallel lines, • intercepting lines, and skew lines • in the cube H G D C E F A B Mr. Chin-Sung Lin**ERHS Math Geometry**Perpendicular Lines and Planes Mr. Chin-Sung Lin**ERHS Math Geometry**Postulates of the Solid Geometry • If two planes intersect, then they intersect in exactly one line B A Mr. Chin-Sung Lin**ERHS Math Geometry**Dihedral Angle • A dihedral angle is the union of two half-planes with a common edge Mr. Chin-Sung Lin**ERHS Math Geometry**• 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 • AC AB and AD AB • The measure of the dihedral angle: • mCAD The Measure of a Dihedral Angle C B D A Mr. Chin-Sung Lin**ERHS Math Geometry**• Perpendicular planes are two planes that intersect to form a right dihedral angle • AC AB, AD AB, and • AC AD (mCAD = 90) • then • m n Perpendicular Planes C m B n D A Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • If a line not in a plane intersects the plane, then it intersects in exactly one point k P A B n Mr. Chin-Sung Lin**ERHS Math Geometry**• A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the intersection of the line and the plane • A plane is perpendicular to a line if the line is perpendicular to the plane • k m, and k n, • then k s A Line is Perpendicular to a Plane k n s m p Mr. Chin-Sung Lin**ERHS Math Geometry**Postulates of the Solid Geometry • At a given point on a line, there are infinitely many lines perpendicular to the given line q p k r n m A Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • 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 k m A P B Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP Prove: k m k m A P B Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • Connect AB • Connect PT and • intersects AB at Q • Make PR = PS k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • Connect RA, SA • SAS • ΔRAP = ΔSAP k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP Prove: k m CPCTC AR = AS k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • Connect RB, SB • SAS • ΔRBP = ΔSBP k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP Prove: k m CPCTC BR = BS k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • SSS • ΔRAB = ΔSAB k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP Prove: k m CPCTC RAB = SAB k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • Connect RQ, SQ • SAS • ΔRAQ = ΔSAQ k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • CPCTC • QR = QS k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • SSS • ΔRPQ = ΔSPQ k R m A Q T P B S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP • Prove: k m • CPCTC • mRPQ = mSPQ • mRPQ + mSPQ = 180 • mRPQ = mSPQ = 90 k R m A Q T B P S Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane Given: Plane p plane q Prove: A line in p is perpendicular to q and a line in q is perpendicular to p C p B A q D Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes If a plane contains a line perpendicular to another plane, then the planes are perpendicular Given: AC in plane p and AC q Prove: p q C p B A q D Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Two planes are perpendicular if and only if one plane contains a line perpendicular to the other C p B A q D Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB p at A Prove: AB is the only line perpendicular to p at A B A p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB p at A Prove: AB is the only line perpendicular to p at A q B C A D p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line Given: Any point P on AB Prove: There is only one plane perpendicular to AB A P B Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line Given: Any point P on AB Prove: There is only one plane perpendicular to AB A Q m n P R B Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes 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 Given: AB p at A and AB AC Prove: AC is in plane p q B C A D p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB p at A, and C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p q B C A p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB p at A, and C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p q B C A D p E Mr. Chin-Sung Lin**ERHS Math Geometry**Parallel Lines and Planes Mr. Chin-Sung Lin**ERHS Math Geometry**Parallel Planes • Parallel planes are planes that have no points in common m n Mr. Chin-Sung Lin**ERHS Math Geometry**A Line is Parallel to a Plane • A line is parallel to a plane if it has no points in common with the plane k m Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • If a plane intersects two parallel planes, then the intersection is two parallel lines p m n Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • If a plane intersects two parallel planes, then the intersection is two parallel lines • Given: Plane p intersects plane m at AB • and plane n at CD, m//n • Prove: AB//CD p B A m D C n Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are parallel • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA//MB q M L B A p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are parallel • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA//MB q N M L B A D C p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are coplanar • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA and MB are coplanar q M L B A p Mr. Chin-Sung Lin**ERHS Math Geometry**Theorems of Parallel Lines & Planes • If two planes are perpendicular to the same line, then they are parallel • Given: Plane p⊥AB at A and q⊥AB at B • Prove: p//q A p B q Mr. Chin-Sung Lin