Lessons 42 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
42
Mutual position of lines in space
Textbook, Part 2: pp. 99–105
GoalKnow the properties of parallel lines in space, the criteria for parallel and skew lines, the properties of a parallelepiped and the angle between skew lines.
New words
parallel lines · parallel to‘g‘ri chiziqlarcriterion for skew lines · ayqashlik alomatiangle between lines · chiziqlar orasidagi burchakdiagonal of a parallelepiped · parallelepiped diagonali
Explanation
Two lines in space are parallel (a ∥ b) if they lie in one plane and do not meet. Through a point not on a given line there is exactly one line parallel to it. If one of two parallel lines meets a plane, so does the other. Two lines parallel to a third line are parallel to each other (the criterion for parallel lines). Criterion for skew lines: if one line lies in a plane and the other meets that plane at a point not on the first line, the lines are skew. The smaller of the adjacent angles formed by two intersecting lines is the angle between them; the angle between skew lines is the angle between intersecting lines parallel to them (it can be drawn from any point, with the same result). Lines with a 90° angle are perpendicular; the angle between parallel lines is taken as 0°. In a parallelepiped opposite faces are equal and parallel, and the four diagonals meet at one point and bisect each other there.
Worked examples
In a cube, lines AC and A₁D are skew. Since A₁D ∥ B₁C, the angle equals ∠ACB₁. The three sides of triangle ACB₁ are face diagonals and are equal, so the angle is 60°.
In a cube AA₁ ∥ BB₁ and CC₁ ∥ BB₁, so by the criterion AA₁ ∥ CC₁. Hence AA₁ and CC₁ lie in one plane and AA₁C₁C is a quadrilateral (in fact a rectangle with sides a and a√2); its diagonals AC₁ and A₁C are diagonals of the cube and bisect each other.
Class activity
Use two pencils (models of lines) and a cardboard box to show two pairs of skew lines and the angle between them (slide one pencil parallel to itself and measure).
Practice
1
State the criterion about two lines parallel to a third line.
They are parallel to each other.
2
In tetrahedron ABCD, K, L, M, N are the midpoints of AB, BC, CD, DA. Which edge are KL and NM parallel to, and why is KLMN a parallelogram?
KL is a midline of triangle ABC and NM of triangle ACD; both are parallel to AC and equal AC/2. So KL ∥ NM and KL = NM, hence KLMN is a parallelogram.
3
In a cube find the angle between lines AB₁ and BC₁ (DC₁ ∥ AB₁).
60°
4
Why does the angle between skew lines not depend on the chosen point?
Pairs of parallel lines drawn through any point have parallel sides, and angles with parallel sides are equal.