Home Page, Courses Menu,, Quarter Schedule, Grading Policy et. al.
.
I. Basic geometric notions
A. Points, lines, and plane
.
1. Points, lines, and planes are basic, but undefined, terms.
2. Space is the set of all points.
3. Collinear points are points that belong to the same line.If three distinct points are collinear, then one of the points is between the other two.
4. Important subsets of lines are segments, half-lines and rays.
a. A line segment consists of two points and all collinear points between them.
b. A half-line is the set of all points on a line in one direction determined by a point on the line
c. A ray is the union of a point and a half-line determined by the point (i.e. a closed half-line).
5. Coplanar points (lines) are points (lines) that lie in the same plan
a. Intersecting lines are two lines with exactly one point in common.
b. Concurrent lines are lines that contain a single common point.
c. Parallel lines are lines with no points in common.
6. Accepted Axioms:
a. A unique line is determined by two distinct points.
b. A unique plane is determined by three non-collinear points. This axiom can be used to prove that any of the following conditions also generates a unique plane.:
(b) Two intersecting lines
(c) Two parallel lines
.
7. Characteristics of planes:
(a) Parallel planes are planes with no points in common.
(b) If the intersection of two distinct planes is not empty, then the intersection is a line.
(c) Planes are parallel if their intersection is empty.
(d) If a line contains two distinct points of a plane, then the line lies in the plane.
8. Skew lines are lines that cannot be contained in the same plane (i.e. cannot be intersecting or parallel).
9 . An angle is the union of two rays with a common endpoint called the vertex.
Standard angles are classified according to size as acute, obtuse, right. Non-standard angles are straight angles (generated by collinear points with a specified point designated as the vertex) and zero degree angles (an angle generated by coincident rays with a common end point).
(a) An acute angle has a measure between 0° and 90°.
(b) A right angle has a measure of 90°.
(c) An obtuse angle has a measure between 90° and 180°.
(d) A straight angle has a measure of 180° (requires a specific point on the line generated by a straight angle to be designated as the vertex).
(e) Two angles are complementary if the sum of their measures is 90°, and supplementary if the sum of their measures is 180°.
(f) Two coplanar angles that have a common vertex and a common side are called adjacent angles if the interiors of the two angles are disjoint.
(g) Two adjacent angles that generate a line are called a linear pair.
10. A dihedral angle is the union of two half-planes and the common line defining the half-planes.
11. Two lines that meet to form a right angle are said to be perpendicular.
.
B. Plane figures
.
1. A closed curve is a curve that, when traced, has the same starting and stopping points and may cross itself at individual points.
2. A curve is simple if, without lifting the pencil, you can draw the curve without retracing any of its points (with the possible exception of its endpoints).
3. A simple closed curve is a simple and closed curve.
a. A simple closed curve separates a plane into three disjoint sets of points: the interior, the exterior, and the curve.
4. A region is the union of the interior of simple closed curve and its boundary (the curve itself.)
(a) A region is convex if, for any two points in the interior of the region, the line segment joining them lies completely in the region
(b) A region is non-convex (concave) if there are two points in the interior of the region such that the line segment joining them intersects the exterior of the region.
5. A polygonal curve is a curve made up of line segments.
6. A polygon is a simple closed polygonal curve (simple definition).
Detailed definition: A polygon is a simple closed curve that is the union of three or more line segments AB, BC, CD, ..., PQ such that A, B, C, D, ..., P, Q are coplanar and distinct, and no three consecutively named points are collinear.
a. Polygons are classified by the number of their sides: triangles (3 sides), quadrilaterals (4), pentagons (5), hexagons (6), heptagons (7), octagons (8), nonagons (9), decagons (10), dodecagons (12), icosagons (20)..., ngons (n sides).
b. A regular polygon is a polygon in which all the corresponding angles are congruent and all the corresponding sides are congruent.
c. A diagonal is any line segment connecting two nonconsecutive vertices of a polygon.
7. A polygonal region is the union of a polygon and its interior.
a. A convex polygon is one that is convex as a region.
b. A concave polygon is a non-convex polygon.
8. Triangles are classified according to the measures of their angles as acute, right or obtuse, and according to the lengths of their sides as scalene, isosceles, or equilateral.
Why are there no other classifications of any triangle by congruency of angles or sides?
9. In general, a quadrilateral can be classified as scalene, or more specifically as a trapezoid, a parallelogram, a rectangle, a square, a rhombus, a kite, or a chevron (non-convex polygon).
10. A circle is a set of points in a plane each of which lies at the same distance from a given point, called the center.
II. Theorems involving angles
A. Supplements of the same angle, or of congruent angles, are congruent.
B. Complements of the same angle, or of congruent angles, are congruent.
C. Vertical angles formed by intersecting lines are congruent. Use Theorem A above to prove this.
D. Use theorems A, B, and C above to prove the following statements.
If any two distinct lines are cut by a transversal, then a pair of corresponding angles, alternate interior angles, or alternate exterior angles are congruent if, and only if, the lines are parallel.
If any two distinct lines are cut by a transversal, then a pair of corresponding interior angles, or corresponding exterior angles are supplementary if, and only if, the lines are parallel.
E. The sum of the measures of the interior angles of a triangle is 180°
The sum of the measures of the interior angles of a quadrilateral is 360°. This pattern generalized to the following statement:
The sum of the measures of the interior angles of any convex polygon
with n sides is
180°n-360°, or (n-2)*180°.
One interior angle of a regular n-gon measures (180n-360)/n, or ((n-2)*180)/ n
F. The sum of the measures of the exterior angles of any polygon is 360°.
IV. Three-dimensional figures (Surfaces)
1. A polyhedron is a simple closed surface in space formed by polygonal regions.
2. Three-dimensional figures with special properties are prisms, pyramids, regular polyhedra, cylinders, cones, and spheres.
(a) A prism is a polyhedron formed by two congruent polygonal regions in parallel planes called bases, parallelogram regions joining the sides of the two polygonal bases so as to form a closed surface.
(b) A pyramid is a polyhedron formed by a simple closed polygonal region making up the base and a point not in the plane of the region, together with the triangular regions that join the point and the edges of the polygonal base. The triangular regions are congruent in a regular pyramid.
(c). Regular polyhedra, The Platonic solids are as follows:
(i) Regular tetrahedron (four triangle faces)
(ii) Regular hexahedron (six triangle faces)
(iii) Regular octahedron (eight triangle faces)
(iv) Regular dodecahedron (twelve triangle faces)
(v) Regular icosahedron (twenty triangle faces)
(d). In general, a cylinder is the surface generated by movement of parallel lines, each of which intersects a closed figure in a plane. Usually we consider the case where the closed figure is a circle, and the bases bound the ends.
(e). A cone consists of segments joining a fixed point not in a given plane to points on a closed curve in the given plane. Usually the closed curve is a circle.
(f). A sphere is the set of all points in space equidistant from a fixed point.
3. Let V represent the number of vertices, E the number of edges, and F the number of faces of a polyhedron. Then Euler's formula states that V+F-E = 2.
4. The table below summarizes classifications and relationships for many geometric concepts. The relationships of congruence and similarity are defined as follows:
Two geometric objects are congruent if they have exactly the same size and shape, that is, either object can replace the other exactly without any gaps or overlaps.
Equivalently: All corresponding angles and linear measures are congruent.
Two geometric objects are similar if they have the same shape, that is, if all the linear measures are proportional, i.e. there exists a fixed ratio between the linear measures of all corresponding parts of the objects.
| Geometric Entity | Classifications | Relationships |
| Lines | Plane, space | Intersecting, skew, perpendicular, parallel, concurrent |
| Curves | Plane, simple, closed, simple closed, polygon, circle | Intersecting, parallel |
| Angles |
Acute, right, obtuse, straight Vertical, adjacent, corresponding, alternate interior, alternate exterior, supplementary, complementary |
|
| Polygons | Concave, convex, triangle, quadrilateral, n-gon | Congruent, similar |
| Triangles |
Angles: acute, right, obtuse |
Congruent, similar |
| Quadri-laterals | Square, rectangle, parallelogram, kite, rhombus, trapezoid | Congruent, similar |
| Solids | Sphere, cylinder, cone, polyhedron | Congruent, similar |
| Poly-hedrons | Prism, pyramid | |
Top of page Return to Geometry Section