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Chapter 4: Parallel Lines, Distances, and Angle Sums

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job trivino

on 13 September 2011

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Transcript of Chapter 4: Parallel Lines, Distances, and Angle Sums

Chapter 4: Parallel Lines, Distances, and Angle Sums 4.2 Distances 4.2A Distances Between Two Geometric Figures The distance between two geometric figures is the straight line segment which is the shortest segment between the figures. 1. The distance between two points, such as P and Q in Fig. 4-23(a), is the line segment PQ between them. 2. The distance between a point and a line, such as P and line AB in (b), is the line segment PQ, the perpendicular from the point to the line. 3. The distance between two parallels, such as line AB and line CD in (c), is the line segment PQ, a perpendicular between the two parallels. 4. The distance between a point and a circle, such as P and circle O in (d), is PQ, the segment of OP between the point and the center of the circle. 5. The distance between two concentric circles, such as two circles whose center is O, is PQ, the segment of the larger radius that lies between two circles, as shown in (e). 4.2B Distance Principles PRINCIPLE 1: If a point is on the perpendicular bisector of a line segment, then it is equidistant from the ends of the line segment. PRINCIPLE 2: If a point is equidistant from the ends of a line segment, then it is on the perpendicular bisector of the line segment. PRINCIPLE 3: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. PRINCIPLE 4: If a point is equidistant from the sides of an angle, then it is on the bisector of an angle. PRINCIPLE 5: Two points each equidistant from the ends of a line segment determine the perpendicular bisector of the line segment. PRINCIPLE 6: The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the vertices of the triangle. PRINCIPLE 7: The bisectors of the angles of a triangle meet in a point which is equidistant from the sides of the triangle. 4.3 Sum of the Measures of the Angles of a Triangle We can prove that the sum of the measures of the angles of a triangle equals 180 degrees by drawing a line through one vertex of the triangle parallel to the side opposite the vertex. 4.3A Interior and Exterior Angles of a Polygon 4.3B Angle-Measure-Sum Principles PRINCIPLE1: The sum of the measures of the angles of a triangle equals the measure of a straight angle. PRINCIPLE2: If two angles of one triangle are congruent respectively to two angles of another triangle, the remaining angles are congruent. PRINCIPLE3: The sum of the measures of the angles of a quadrilateral equals 360 degrees. PRINCIPLE4: The measure of each exterior angle of a triangle equals the sum of the measures of its two nonadjacent interior angles. PRINCIPLE5: The sum of the measures of the exterior angles of a triangle 360 degrees. PRINCIPLE6: The measure of each of an equilateral triangle 60 degrees. PRINCIPLE7: The acute angles of a right triangle are complementary. PRINCIPLE8: The measures of each acute angle of an isosceles right triangle equals 45 degrees. PRINCIPLE9: A triangle can have no more than 1 right angle. PRINCIPLE10: A triangle can have no more than one obtuse angle. PRINCIPLE11: Two angles are supplementary if their sides are respectively perpendicular to each other.
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