Parallel Lines: Two lines in the same plane are parallel if they do not intersect or are the same.
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1 Section 2.3: Lines and Angles Plane: infinitely large flat surface Line: extends infinitely in two directions Collinear Points: points that lie on the same line. Parallel Lines: Two lines in the same plane are parallel if they do not intersect or are the same. Skew Lines: Two lines that do not intersect and are not parallel. Concurrent Lines: Three or more lines that contain the same point. Properties of Points and Lines: * For each pair of points A and B (A B) in the plane, there is a unique line AB containing them. * The distance between points A and B is the nonnegative difference of the real numbers a and b to which A and B correspond. The distance is written AB or BA. (a and b are called coordinates of A and B on AB). * If a point P is not on a line l, there is a unique line m, m l, such that P is on m and m is parallel to l, written m l.
2 2 SECTION 2.3: LINES AND ANGLES Angle: An angle is the union of two line segments (or two rays) with a common endpoint, called the vertex. A B C α Adjacent angles: Adjacent angles are two angles that share a vertex, have a common side, but whose interiors do not intersect. A B D C Classification of Angles According to Measurement: * acute angle: angle measuring less than 90 * right angle: angle measuring 90 * obtuse angle: angle measuring more than 90 but less than 80 * straight angle: angle measuring 80 * reflex angle: angle measuring more than 80 Classification of Triangles According to Angles: * right triangle: triangle with a right angle. * obtuse triangle: triangle with an obtuse angle. * acute triangle: triangle with all acute angles. * equiangular triangle: triangle with all angles the same measurement. IMPORTANT: The sum of the measures of the angles of a triangle is 80.
3 SECTION 2.3: LINES AND ANGLES 3 Vertical Angles: Opposite angles formed by two intersecting lines are called vertical angles. IMPORTANT: Vertical angles always have the same measurement In the above figure, and 3 are vertical angles; 2 and 4 are vertical angles. Complementary Angles: Two angles whose sum is 90 are called complementary angles. If A and B are complementary angles, then A is the complement of B and B is the complement of A. 2 In the above figure, and 2 are complementary angles. Supplementary Angles: Two angles whose sum is 80 are called supplementary angles. If A and B are supplementary angles, then A is the supplement of B and B is the supplement of A. 2 In the above figure, and 2 are supplementary angles.
4 4 SECTION 2.3: LINES AND ANGLES Example : Find the measure of each marked angle. (a) (x - 34) (4x +9) (b) (3x + 25) (6x - 7) (c) (3x-0) (6x+)
5 SECTION 2.3: LINES AND ANGLES 5 ANGLES ASSOCIATED WITH PARALLEL LINES: Corresponding Angles have the same location relative to lines l, m and transversal t. IMPORTANT: l m if and only if corresponding angles formed by l, m, and t are congruent. In Figure A-, and 5 are corresponding angles. The following pairs are also corresponding angles: 2 and 6; 3 and 7; 4 and 8. t l m Figure A- Alternate Interior Angles are nonadjacent angles formed by lines l, m, and transversal t, the union of whose interiors contain the region between l and m. IMPORTANT: l m if and only if alternate interior angles formed by l, m and t are congruent. In Figure A-, 3 and 6 are alternate interior angles. Likewise, 4 and 5 are also alternate interior angles.
6 6 SECTION 2.3: LINES AND ANGLES Alternate Exterior Angles are angles on the outer sides of two lines cut by a transversal, but on opposite sides of the transversal. IMPORTANT: l m if and only if alternate exterior angles formed by l, m and t are congruent. In Figure A-, 2 and 7 are alternate exterior angles. Similarly, and 8 are alternate exterior angles. Interior Angles on the same side of the transversal are interior angles whose interiors are the same. IMPORTANT: l m if and only if the interior angles on the same side of the transversal are supplementary. In Figure A-, 3 and 5, as well as 4 and 6, are interior angles on the same side of the transversal. Example 2: In the diagram below, l m and r s. Find the measurement of each numbered angle l m r s
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