Download Math 3372-College Geometry

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
Contents
1
Review of Topics in Secondary School Geometry
1.1
2
3
Triangles and Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exploring Geometry
1
1
4
2.1
A few examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Discovery via the Computer . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Foundations of Geometry 1: Points, Lines, Segments, Angles
7
3.1
An Introduction to Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Axioms, Axiomatic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.3
Incidence Axioms for Geometry . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.4
Distance, Ruler Postulate, Segments, Rays, and Angles . . . . . . . . . . . .
11
3.5
Angle measure and the Protractor postulate . . . . . . . . . . . . . . . . . .
14
ii
List of Figures
iii
List of Tables
iv
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
∆ABC ∼
= ∆DEF
This means:
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 1800 .
2 Remark: The position of the side in one triangle must be the counterpart of the side in
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Chapter 1.
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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.
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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 = 900 .
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.
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Chapter 1.
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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.
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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.
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