Geometry Chapter 2
regular polygrams
When the polygon is regular, the resulting polygram is also regular—that is, the interior acute angles are congruent, the interior reflex angles are congruent, and all sides are congruent.
Riemann Postulate
through a point not on a line, there are no lines parallel to the given line. (Spherical Geometry)
The total number of diagonals D in a polygon of n sides is given by the formula
D = n(n-3)/2
Scalene Triangle
Has 0 sides that are congruent
Triangle
Is the union of three line segments that are determined by three non-collinear points
Is there a relationship between Conditionals and their contrapositives?
"If P, then Q" and "If not Q, then not P" are equivalent.
Theorems related to parallel lines
1. If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 2. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent 3. If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. 4. If two parallel lines are cut by a transversal, then the pairs of exterior angles on the same side of the transversal are supplementary.
A diagonal of a polygon
A diagonal of a polygon is a line segment that joins two nonconsecutive vertices.
Point Symmetry
A figure has symmetry with respect to point P if for every point M on the figure, there is a second point N on the figure for which point P is the midpoint of MN.
Non-Euclidean Geometry
A non-Euclidean geometry is a geometry characterized by the existence of at least one contradiction of a Euclidean geometry postulate
Polygon
A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoints.
Polygram
A polygram is the star-shaped figure that results when the sides of convex polygons with five or more sides are extended
Regular Polygon
A regular polygon is a polygon that is both equilateral and equiangular.
hyperbolic paraboloid
A saddle-like surface
Corollary
A theorem that follows directly from a previous theorem is known as a corollary of that theorem. Corollaries, like theorems, must be proved before they can be used. These proofs are often brief, but they depend on the related theorem. E.g., -Each angle of an equiangular triangle measures 60 degrees. -The acute angles of a right triangle are complementary -If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
Transversal Lines
A transversal is a line that intersects two (or more) other lines at distinct points; all of the lines lie in the same plane.
Interior Angles and Exterior Angles
A transversal is a line that intersects two (or more) other lines at distinct points; all of the lines lie in the same plane. In Figure 2.6, t is a transversal for lines r and s. Angles that are formed between r and s are interior angles; those outside r and s are exterior angles. Relative to Figure 2.6, we have Interior angles: ∠3, ∠4, ∠5, ∠6 Exterior angles: ∠1, ∠2, ∠7, ∠8
Perpendicular Lines and a theorem
By definition, two lines (or segments or rays) are perpendicular if they meet to form congruent adjacent angles. Using this definition, we proved the theorem stating that "perpendicular lines meet to form right angles." From a point not on a given line, there is exactly one line perpendicular to the given line. The term perpendicular includes line-ray, line-plane, and plane-plane relationships
For the conditional statement that follows, give the converse, the inverse, and the contrapositive. Then classify each as true or false. If two angles are vertical angles, then they are congruent angles
CONVERSE: If two angles are congruent angles, then they are vertical angles. (false) INVERSE: If two angles are not vertical angles, then they are not congruent angles. (false) CONTRAPOSITIVE: If two angles are not congruent angles, then they are not vertical angles. (true)
Corresponding Angles
Consider the angles in that are formed when lines are cut by a transversal. Two angles that lie in the same relative positions (such as above and left) are called corresponding angles for these lines.
THE LAW OF NEGATIVE INFERENCE (CONTRAPOSITION)
Consider the following circumstances, and accept each premise as true: 1. If Matt cleans his room, then he will go to the movie. (P S Q) 2. Matt does not get to go to the movie. (~Q) What can you conclude? You should have deduced that Matt did not clean his room; if he had, he would have gone to the movie. This "backdoor" reasoning is based on the fact that the truth of P S Q implies the truth of ~Q S ~P.
Strategy for Proof (The First Line of an Indirect Proof)
General Rule: The first statement of an indirect proof is generally "Suppose/Assume the opposite of the Prove statement."
Isosceles Triangle
Has 2 sides that are congruent
Equiangular Triangle
Has 3 angles that are equal
Equilateral Triangle
Has three sides that are congruent
great circle
If a line segment on the surface of the sphere is extended to form a line, it becomes a great circle (like the equator of the earth). Each line in this geometry, known as spherical geometry, is the intersection of a plane containing the center of the sphere with the sphere.
Transversal with corresponding, alternate interior/exterior theorems
If two lines are cut by a transversal so that two corresponding angles are congruent, then these lines are parallel. If two lines are cut by a transversal so that two alternate exterior angles are congruent, then these lines are parallel.
Postulate 11 (parallel lines cut by transversals)
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Line Symmetry
In the figure below, rectangle ABCD is said to have symmetry with respect to line because each point to the left of the line of symmetry or axis of symmetry has a corresponding point to the right; for instance, X and Y are corresponding points. A figure has symmetry with respect to a line if for every point X on the figure, there is a second point Y on the figure for which is the perpendicular bisector of XY.
Indirect Proof
Indirect proof is synonymous with proof by contradiction. A keyword signalling that you should consider indirect proof is the word 'not'. Usually, when you are asked to prove that a given statement is NOT true, you can use indirect proof by assuming the statement is true and arriving at a contridiction.The idea behind the indirect method is that if what you assumed creates a contradiction, the opposite of your initial assumption is the truth. Whereas the Law of Detachment characterizes the method of "direct proof" found in preceding sections, the Law of Negative Inference char- acterizes the method of proof known as indirect proof.
Concave and Convex Polygons
Most polygons considered in this textbook are convex; the angle measures of convex polygons are between 0 and 180. Some convex polygons are shown in Figure 2.28; those in Figure 2.29 are concave. A line segment joining two points of a concave polygon can contain points in the exterior of the polygon. Thus, a concave polygon always has at least one reflex angle.
Parallel Lines
Parallel lines are lines in the same plane that do not intersect.
Method For Indirect Proof (Step by step strategy)
Prove the statement P -> Q or to complete the proof problem of the form Given: P Prove: Q by the indirect method, use the following steps: 1. Suppose that ~Q is true. 2. Reason from the supposition until you reach a contradiction. 3. Note that the supposition claiming that ~Q is true must be false and that Q must therefore be true. Step 3 completes the proof
The measure E of each exterior angle of a regular polygon or equiangular polygon of n sides is E =
The measure E of each exterior angle of a regular polygon or equiangular polygon of n sides is E = 360/n
The measure I of each interior angle of a regular polygon or equiangular polygon of n sides is
The measure I of each interior angle of a regular polygon or equiangular polygon of n sides is I= (n-2)*180/n
Corollary of exterior angle measures and their relationship to the interior angles of a triangle
The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles.
The sum of the measures of interior angles of a polygon.
The sum S of the measures of the interior angles of a polygon with n sides is given by S = (n - 2) * 180. Note that n > 2 for any polygon.
Sum of measures of exterior angles of a polygon (one at each vertex)?
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.
Sum of the measures of a quadrilateral?
The sum of the measures of the four interior angles of a quadrilateral is 360°.
The sum of Interior angles in a triangle
The sum of the measures of the three interior angles of a triangle is 180°. This is proved through the use of an auxiliary (or helping) line. When an auxiliary line is added to the drawing for a proof, a justification must be given for the existence of that line
Parallel Postulate
Through a point not on a line, exactly one line is parallel to the given line.
How many diagonals do the various polygons have?
Triangle 3 sides 0 diagonals Quadrilateral 4 sides 2 diagonals Pentagon 5 sides 5 diagonals Hexagon 6 sides 9 diagonals
Categories of Polygons
Triangle- 3 sides Quadrilateral- 4 Pentagon- 5 Hexagon- 6 Heptagon- 7 Octagon- 8 Nonagon- 9 Decagon- 10
Alternate Angles (exterior and interior)
Two interior angles that have different vertices and lie on opposite sides of the transversal are alternate interior angles. Two exterior angles that have different vertices and lie on opposite sides of the transversal are alternate exterior angles. Both types of alternate angles must occur in pairs; in Figure 2.6, we have:
Underdetermined and overdetermined Auxiliary Lines
When an auxiliary line is introduced into a proof, the original drawing is redrawn for the sake of clarity. Each auxiliary figure must be determined, but not underdetermined or overdetermined. A figure is underdetermined when more than one figure is possible. On the other extreme, a figure is overdetermined when it is impossible for the drawing to include all conditions described
Exterior Angle of a Triangle
When the sides of a triangle are extended, each angle that is formed by a side and an extension of the adjacent side is an exterior angle of the triangle
Universe and Disjoint Definition
With Venn Diagrams, the set of all objects under consideration is called the universe. If P = {all polygons} is the universe, then we can describe sets T = {triangles} and Q = {quadrilaterals} as subsets that lie within the universe P. Sets T and Q are described as a disjoint because they have no elements in common.
Transformations
transformations preserve lengths and angle measures and thus lead to a second figure that is congruent to the given figure . The types of transformations included are (1) the slide or translation, (2) the reflection, and (3) the rotation. Each of these types of transformations is also called an isometry, which translates to "same measure."