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Transcript
9/24/2010 Parallel Lines and Planes 3-1 Definitions Parallel Lines coplanar lines that do not intersect Parallel Planes Parallel planes that do not intersect Skew Lines noncoplanar lines THEOREM 3-1 If two parallel planes are cut by a third plane, then the lines of intersection are parallel. 1 9/24/2010 Transversal is a line that intersects each of two other coplanar lines in different points to produce interior and exterior angles Interior Angles 2 1 4 Interior Angles The angles formed between the two coplanar lines and the transversal Exterior Angles The angles formed outside the two coplanar lines and the transversal 3 Exterior Angles 8 6 5 7 ALTERNATE INTERIOR ANGLES two nonadjacent interior angles on opposite sides of a transversal 2 9/24/2010 Alternate Interior Angles 2 1 4 3 Alternate Exterior Angles 8 6 5 7 Same-Side Interior Angles 2 1 4 3 ALTERNATE EXTERIOR ANGLES two nonadjacent exterior angles on opposite sides of the transversal Same-Side Interior Angles two interior angles on the same side of the transversal Corresponding Angles two angles in corresponding positions relative to two lines cut by a transversal 3 9/24/2010 Corresponding Angles 6 2 5 1 4 8 3 7 3-2 Properties of Parallel Lines Postulate 10 If two parallel lines are cut by a transversal, then corresponding angles are congruent. THEOREM 3-3 If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. THEOREM 3-2 If two parallel lines are cut by a transversal, then alternate interior angles are congruent. THEOREM 3-4 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also. 4 9/24/2010 3-3 Proving Lines Parallel Postulate 11 If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel THEOREM 3-5 If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. THEOREM 3-6 If two lines are cut by a transversal and sameside interior angles are supplementary, then the lines are parallel. THEOREM 3-7 In a plane two lines perpendicular to the same line are parallel. THEOREM 3-8 Through a point outside a line, there is exactly one line parallel to the given line. 5 9/24/2010 THEOREM 3-9 Through a point outside a line, there is exactly one line perpendicular to the given line. THEOREM 3-10 Two lines parallel to a third line are parallel to each other. Ways to Prove Two Lines Parallel 1. Show that a pair of corresponding angles 2. 3. 4. 5. are congruent. Show that a pair of alternate interior angles are congruent Show that a pair of same-side interior angles are supplementary. In a plane show that both lines are ⊥ to a third line. Show that both lines are || to a third line Triangle – is a figure formed by the segments that join three noncollinear points 3-4 Angles of a Triangle Vertex – point of a triangle Side – segment of a triangle 6 9/24/2010 Scalene triangle – is a Isosceles Triangle – is triangle with all three sides of different length. a triangle with at least two legs of equal length and a third side called the base and Angles at the base are called base angles and the third angle is the vertex angle Equilateral triangle – is a triangle with three sides of equal length Acute triangle – is a Obtuse triangle – is a triangle with three acute angle (<90°) triangle with one obtuse angle (>90°) 7 9/24/2010 Right triangle – is a triangle with one right angle (90°) Auxillary line – is a line (ray or segment) added to a diagram to help in a proof. Corollary Equiangular triangle – is a triangle with three angles of equal measure. THEOREM 3-11 The sum of the measures of the angles of a triangle is 180° Corollary 1 A statement that can If two angles of one easily be proved by applying a theorem triangle are congruent to two angles of another triangle, then the third angles are congruent. 8 9/24/2010 Corollary 2 Corollary 3 Each angle of an In a triangle, there can be equiangular triangle has measure 60°. at most one right angle or obtuse angle. Corollary 4 THEOREM 3-12 The acute angles of a right The measure of an exterior triangle are complementary. angle of a triangle equals the sum of the measures of the two remote interior angles. Polygon – is a closed 3-5 Angles of a Polygon plane figure that is formed by joining three or more coplanar segments at their endpoints, and 9 9/24/2010 Each segment of the polygon is called a side, and the point where two sides meet is called a vertex, and Convex polygon - is a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon. The angles determined by the sides are called interior angles. Diagonal - a segment of a polygon that joins two nonconsecutive vertices. THEOREM 3-14 THEOREM 3-13 The sum of the measures of the angles of a convex polygon with n sides is (n-2)180° The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360° 10 9/24/2010 Inductive Reasoning 3-6 Inductive Reasoning Conclusion based on several past observations Conclusion is probably true, but not necessarily true. Deductive Reasoning Conclusion based on accepted statements Conclusion must be true if hypotheses are true. THE END 11