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    4 Parallel Planes and Lines

    Uploaded by

    BAHRI
    This document discusses properties and theorems related to parallel planes and lines in geometry. It defines what makes planes and lines parallel and explores their relationships. Some key points covered include: if a line is parallel to a plane, it is parallel to the intersection of any plane through it with the given plane. Also, through a point not in a plane there is one and only one plane parallel to that plane. Finally, if two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.

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    © All Rights Reserved
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    4 Parallel Planes and Lines

    Uploaded by

    BAHRI
    74 views9 pages
    This document discusses properties and theorems related to parallel planes and lines in geometry. It defines what makes planes and lines parallel and explores their relationships. Some key points covered include: if a line is parallel to a plane, it is parallel to the intersection of any plane through it with the given plane. Also, through a point not in a plane there is one and only one plane parallel to that plane. Finally, if two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.

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    geometry

    Original Title

    4 Parallel planes and lines

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    This document discusses properties and theorems related to parallel planes and lines in geometry. It defines what makes planes and lines parallel and explores their relationships. Some key points covered include: if a line is parallel to a plane, it is parallel to the intersection of any plane through it with the given plane. Also, through a point not in a plane there is one and only one plane parallel to that plane. Finally, if two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    74 views9 pages

    4 Parallel Planes and Lines

    Uploaded by

    BAHRI
    This document discusses properties and theorems related to parallel planes and lines in geometry. It defines what makes planes and lines parallel and explores their relationships. Some key points covered include: if a line is parallel to a plane, it is parallel to the intersection of any plane through it with the given plane. Also, through a point not in a plane there is one and only one plane parallel to that plane. Finally, if two straight lines are cut by three parallel planes, the intercepted segments on one line are proportional to the corresponding segments on the other.

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    PROPERTIES OF THE PLANE:

    Parallel planes and lines


    AN Cahyono
    Parallel planes and lines
    • Parallel planes
    ….

    • Line parallel to a plane


    ….

    • Intercepted segments
    ….
    Line parallel to a plane
    • Theorem VIII. If a straight line is parallel to a given plane, it is parallel to the
    intersection of any plane through it with the given plane.
    Proof: ….

    Corollary 1. if a line outside a plane is parallel to some line in the plane, then
    the first line is parallel to the plane
    Corollary 2. if a line is parallel to a plane, then through any point there is a
    line in the plane parallel to the given line
    Corollary 3. The intersections of a plane with two parallel planes are parallel
    lines
    Planes perpendicular to a line are parallel
    • Theorem IX. If each of two planes is perpendicular to the same line, they
    are parallel; and conversely, if one of two parallel planes is perpendicular to
    a line, the other is also.
    Proof: ….

    Corollary 1. Parallel line-segments included between parallel planes are


    equal.
    Corollary 2. If a line is perpendicular to one of two parallel planes, it is
    perpendicular to the other also.
    Corollary 3. Two planes each parallel to a third plane are parallel to each
    other.
    Parallel planes
    • Theorem X. If a plane is parallel to each of two intersecting lines, it is
    parallel to the plane of these lines.
    Proof: ….

    • Theorem XI. Through a point not in a plane there is one and only one
    plane parallel to this plane.
    Proof: ….
    Planes parallel to given lines
    • Theorem XII. Through one of two skew lines there is one and only one
    plane parallel to the other line.
    Proof: ….

    • Theorem XIII. Through a point outside of each of two non-parallel


    lines there is one and only one plane parallel to both of these lines.
    Proof: ….
    Angles whose sides are parallel
    • Theorem XIV. If two intersecting lines in one plane are parallel,
    respectively, to two intersecting lines in another plane, then the two
    planes are parallel, and then the corresponding angles formed by the
    lines are equal.
    Proof: ….

    Corollary. Two angles in space whose sides are parallel each to each are
    either equal or supplementary.
    Parallel planes intercept proportional
    segments
    • Theorem XV. If two straight lines are cut by three parallel planes, the
    intercepted segments on one line are proportional to the
    corresponding segments on the other.
    Proof: ….

    Corollary. Parallel planes which intercept equal segments on any


    transversal line intercept equal segments on every transversal line.
    Thank you

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