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GEOMETRY Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ M3 Period: _____Date______________ Lesson 2: Points, Lines and Planes Learning Target: I can describe the properties of points, lines, and planes Opening Exercise: Fill in the blanks. Draw any missing pictures in the 2nd column. Element Description A point is a position in space, and has no _____________________ (actual size). A point is usually named with a capital _________________. In the coordinate plane, a point is named by an ordered pair, (π₯, π¦). A Line has no thickness but its length extends in one dimension and goes on forever in both β‘ . directions. A line is named by a single lowercase letter, l, or by any two points on the line, π΄π΅ A _________ is a part of a line that starts at an endpoint and extends forever in one direction. A ray is named by the endpoint followed by a point on the ray, such as π΄π΅ Opposite rays are 2 rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight ________ and/or a straight ________ (180°). A Line segment is part of a line and consists of two points and all the points between. A Μ Μ Μ Μ . segment is named by its two ___________________, such as πΆπ· A plane has no thickness but extends indefinitely in all directions. A plane is named by a single __________ (plane V ) or by three non-___________________ points (plane π ππ). Collinear Points are points that lie on the same line, such as ____ and ____ in the example at right. Coplanar points are points that lie in the ____________ plane such as π΄, π΅, πππ πΆ in the example at right. Parallel lines two lines that do not ___________________ . Intersecting lines are lines that intersect at a _________________ . Skew lines two _____-____________ lines that do not ________________ . Example GEOMETRY Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ M3 Period: _____Date______________ Example 1. 1. Name a plane. ______________________________ 2. Name a segment. ______________________________ 3. Name a line. ______________________________ 4. Name three collinear points. ________________________ 5. Name three non collinear points. _____________________ 6. Name the intersection of a line and a segment not on the line. _________________ Example 2. 1. Name a pair of opposite rays. _________________________________ 2. Name the points that determine plane πΉ. _______________________ 3. Name the point at which line π intersect plane πΉ. ________________ 4. Name two lines in plane πΉ that intersect line π. __________________ 5. Name a line in plane πΉ that does not intersect line π. ______________ Theorems related to lines and planes 1. One Line Perpendicular to One Plane Through a given point on a line passes only one only one plane perpendicular to a given line. Through any point on a plane passes one and only one line perpendicular to that plane. 2. One Line Perpendicular to Two or More Planes Two planes perpendicular to the same line are parallel. πππππ π1 and πππππ π are perpendicular to β‘πΆπ·, therefore πππππ π1 β₯ πππππ π GEOMETRY Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ M3 Period: _____Date______________ 3. Two Lines Perpendicular to One Plane β‘ and Two lines perpendicular to a plane are parallel and coplanar (π΄π΅ β‘ are perpendicular to plane q, therefore π΄π΅ β‘ β₯ πΆπ· β‘ and they are πΆπ· coplanar) 4. Two Perpendicular Planes Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane. 5. Two Perpendicular Lines and Planes at a Point 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 given plane. 6. One Line Perpendicular to One Plane: Revisited If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane. 7. Parallel Planes Intersecting with Another Plane If a plane intersects two parallel planes, then the intersection is two parallel lines. Example 3. Indicate whether each statement is always true (A), sometimes true (S), or never true (N). a. ________ If two lines are perpendicular to the same plane, the lines are parallel. b. ________ Two planes can intersect in a point. c. Two lines parallel to the same plane are perpendicular to each other. ________ d. ________ If a line meets a plane in one point, then it must pass through the plane. e. ________ Skew lines can lie in the same plane. f. If two lines are parallel to the same plane, the lines are parallel. ________ g. ________ If two planes are parallel to the same line, they are parallel to each other. h. ________ If two lines do not intersect, they are parallel. GEOMETRY Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ M3 Period: _____Date______________ Lesson 2: Points, Lines and Planes Classwork 1. Answer all the questions using the diagram on the right a. Name the plane. ______________________ b. Name a segment. ______________________ c. Name a line. ______________________ d. Name three collinear points. ______________________ e. Name three non collinear points. ______________________ f. Name the intersection of a line and a segment not on the line. g. Name a pair of opposite rays. ______________________ ______________________ h. Name the points that determine plane R. ______________________ i. Name the point at with line β‘π½πΎ intersects plane R. ______________________ j. Name two lines in plane R that intersect point π. ______________________ k. Name a line in plane R that does not intersect point π·. 2. If is contained in plane P, and 1) 2) 3) 4) ______________________ is perpendicular to plane R, which statement is true? is parallel to plane R. Plane P is parallel to plane R. is perpendicular to plane P. Plane P is perpendicular to plane R. 3. As shown in the diagram below, , , and intersect at π΄. is contained in plane R, are contained in plane S, and and Which fact is sufficient to show that planes R and S are perpendicular? 1) 2) 3) 4) 4. Point π΄ is not contained in plane B. How many lines can be drawn through point π΄ that will be perpendicular to plane B ? a. One b. Two c. Zero d. Infinity GEOMETRY Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ M3 Period: _____Date______________ 5. In the diagram at right, point πΎ is in plane P. How many lines can be drawn through πΎ, perpendicular to plane P ? a. One b. Two c. Zero d. Infinity 6. In plane P, lines π and π intersect at point π΄. If line π is perpendicular to line π and line π at point π΄, then line π is contained in plane P 1) parallel to plane P 2) perpendicular to plane P 3) skew to plane P 4) 7. Lines a and b intersect at point P. Line c passes through P and is perpendicular to the plane containing lines a and b. Which statement must be true? 1) Lines a, b, and c are coplanar. 2) Line a is perpendicular to line b. 3) Line c is perpendicular to both line a and line b. 4) Line c is perpendicular to line a or line b, but not both. 8. Lines and intersect at point E. Line m is perpendicular to lines Which statement is always true? 1) Lines and are perpendicular. 2) Line m is parallel to the plane determined by lines and . 3) Line m is perpendicular to the plane determined by lines and . 4) Line m is coplanar with lines and . and at point E. GEOMETRY NYS COMMON CORE MATHEMATICS CURRICULUM Name________________________ Lesson 2 M3 Period: _____Date______________