Math 3372-College Geometry

Transcription

1 Math 3372-College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang

2 Contents 1 Review of Topics in Secondary School Geometry Triangles and Congruence Exploring Geometry A few examples Discovery via the Computer Foundations of Geometry 1: Points, Lines, Segments, Angles An Introduction to Proof Axioms, Axiomatic Systems Incidence Axioms for Geometry Distance, Ruler Postulate, Segments, Rays, and Angles Angle measure and the Protractor postulate ii

3 List of Figures iii

4 List of Tables iv

5 Chapter 1 Review of Topics in Secondary School Geometry 1.1 Triangles and Congruence Triangle: vertices, sides, angles. For example: ABC Congruence: Two triangles are said to be congruent if the pairs of corresponding sides and angles in the two triangles are congruent, ie., have the same measure. We write This means: ABC = DEF In general, CPCF: Corresponding parts of congruent figures are congruent, Such as two congruent polygons. Congruence Postulates: SAS, ASA, SSS, AAS SAS: If two triangles have two sides and the included angle of one congruent, respectively, to the corresponding two sides and included angle of the other, the triangles are congruent. ASA: If two triangles have two angles and the included side of one congruent, respectively, to the corresponding two angles and included side of the other, the triangles are congruent. SSS: If two triangles have their corresponding sides congruent, the triangles are congruent. AAS: If two triangles have angles and a side opposite one of them in one triangle congruent, respectively, to the corresponding two angles and opposite side in the second triangle, the triangles are congruent. Proof: by using ASA and and use the Angle-Sum Theorem in Euclidean geometry: the sum of the measures of the interior angles of a triangle equals Remark: The position of the side in one triangle must be the counterpart of the side in 1

6 Yi Wang Chapter 1. 2 the other triangle to apply AAS. Remark:: There is no SSA. (except for right triangles) Isosceles triangles: a triangle with two congruent sides Isosceles Triangle Theorem ABC with AB = AC, then B = C. the converse is also true. i.e., If two sides of a triangle are congruent, the angles opposite those sides are congruent. Conversely, if two angles are congruent, the sides opposite are congruent. Proof: ( ) By constructing the midpoint M of the base and using SSS. ( ): By constructing the angle bisector and using SAS. Example 1.1 In Figure 1.1 it is given that W L = KT and W LT = KT L. Show that LK = W T and KLT = W T L. Example 1.2 A truss is built in the shape shown in Figure 1.2, where AB = AC, m BDC = 90, and m DBC = m DCB = 45. Find the sum of the measures of the angles marked 1,2, and 3. Generalize this result. For the generalization: We don t need AB = AC and m DBC = m DCB = 45, But we need D = Right triangle: is a triangle with a right angle. The Hypotenuse is the side opposite the right angle, and the legs are the sides adjacent to the right angle. HA, LA, and HL congruence criteria HA: Two right triangles are congruent if they have an acute angle and hypotenuse of one congruent, respectively, to an acute angle and hypotenuse of the other. see the following figure. LA: If under some correspondence two right triangles have a leg and acute angle of one congruent, respectively, to the corresponding leg and acute angle of the other, the triangles are congruent. HL: If two right triangles have a leg and hypotenuse congruent, respectively, to the corresponding leg and hypotenuse of the other, the triangles are congruent. Note: this criteria is special for right triangles. Proof: Need special construction: see the following figure. Extend segment Y Z to point W such that ZW = BC.

7 Yi Wang Chapter 1. 3 Circumcircle and Incircle of a Triangle; Locus Locus: The locus of a point is the path or set of points that is determined by that point when it satisfies certain given properties. Theorem 1.3 (1) The locus of a point equidistant from the end points of a line segment is the perpendicular bisector of that line segment. (2) The locus of a point equidistant from the sides of an angle is the bisector of that angle. Corrolary 1.4 The perpendicular bisectors of the sides of any triangle are concurrent in a point O that is equidistant from the three vertices, hence is the center of a circle (circumcircle )passing through the vertices. Proof: Trivial. Corrolary 1.5 The bisectors of the angles of any triangle are concurrent in a point I that is equidistant from the three sides of the triangle, hence is the center of a circle (incircle ) tangent to the sides. Proof: Trivial.