Wednesday, October 22, 2008

Geometry Chapter 4 - Congruence of line segments, angles, and triangles

Chapter 4 - Congruence of line segments, angles, and triangles - notes:
4-1 Postulates of lines, line segments, and angles –
Postulates:

4.1 A line segment can be extended to any length in either direction.
4.2 Through two given points, one and only one line can be drawn.
4.3 Two lines cannot intersect in more than one point.
4.4 One and only one circle can be drawn with any given point as it's center and the length of any given line segment as the radius.
4.5 At a given point on a given line, one and only one perpendicular can be drawn to the line.
4.6 From a given point not on a given line, one and only one perpendicular can be drawn to the line.
4.7 For any two distinct points, there is only one positive real number that is the length of the line segment joining the two endpoints.
4.8 The shortest distance between 2 points is the length of the line segment joining these two points.
4.9 A line segment has one and only one midpoint.
5.0 An angle has one and only one bisector.

4.2 Using Postulates and Definitions in Proofs:
We use the laws of logic to combine definitions and postulates to prove a theorem.
We will assume unless otherwise stated, that lines that appear to be straight lines in a figure actually are straight lines and that points that appear on the line actually are.

Example: Given: and AB = CD.
Prove: AC = BD
Statements Reasons
1. AB = CD 1. Given
2. BC = BC 2. Reflexive Postulate
3. AB + BC = CD + BC 3. Addition Postulate
4. AB + BC = AC; CD + BC = BD 4. partition postulate
5. AC = BD 5. Substitution postulate

Theorem 4.1 If 2 angles are right angles, then they are congruent.
Theorem 4.2 If two angles are straight angles, then they are congruent.

Definitions involving Pairs of Angles:

Definition:

Adjacent Angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common.
Definition: Complementary angles are two angles, the sum of whose degree measure is 90 degrees.
Definition: Supplementary angles are two angles, the sum of whose degree measure is 180 degrees.
Theorem 4.3 If two angles are complements of the same angle or congruent angles, then the two angles are congruent.
Theorem 4.4 If two angle are congruent, then their complements are congruent.
Theorem 4.5 If two angles are supplements of the same angle, then they are congruent.
Theorem 4.6 If two angles are congruent, then their supplements are congruent.
If two angles are congruent, then they have the same supplement.

Definition: A linear pair of angles are two adjacent angles whose sum is a straight angle.
Theorem 4.7 If two angles form a linear pair, then they are supplementary.
Theorem 4.8 If two lines intersect to form congruent adjacent angles, then they are perpendicular.

Definition: Vertical angles are two angles in which the sides of one angle are opposite rays to the sides of the second angle.
Theorem 4.9: If two lines intersect, then the vertical angles are congruent.

How to present a proof in geometry using deductive reasoning.
1. As an aid, draw a figure that pictures the data of the theorem or the problem. Use letters to label points in the figure.
2. State the given, which is the hypothesis of the theorem, in terms of the figure.
3. State the Prove, which is the conclusion of the theorem, in terms of the figure.
4. Present the proof, which is a series of logical arguments used in the demonstration. Each step in the proof should consist of a statement about the figure. Each statement should be justified by the given, a definition, a postulate, or a previously proved theorem. The proof may be presented in a two-column format or in a paragraph form.

4.4 Congruent Polygons and Corresponding Parts
Corresponding Parts of Congruent Polygons

Two Polygons are congruent if and only if there is a one-to-one correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent.
Therefore: Corresponding parts of congruent polygons are congruent.

Congruent Triangles:
C
orresponding parts of congruent triangles are congruent (CPCTC).

if two triangles are congruent, then the corresponding parts are congruent.

Postulates:
Reflexive Postulate:
any geometric figure is congruent to itself.
Symmetric Postulate: a congruence may be expressed in either order.
Transitive Postulate: Two geometric figures congruent to the same geometric figure are congruent to each other.

4.5 - 4.7 Proving Triangles Congruent using
5 ways to show Triangle Congruence:

1. SAS = SAS Congruence Postulate - if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.Example: Given ΔABC and ΔDEF,if AB = DE, BC = EF and angle B = angle E, then ΔABC = ΔDEF.

2. SSS = SSS Congruence Postulate - if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.Example: Given ΔABC and ΔDEF,if AB = DE, BC = EF and AC = DF, then ΔABC = ΔDEF.

3. ASA = ASA Congruence Postulate - if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.Example: Given ΔABC and ΔDEF,if angle A = angle D, AB = DE, and angle B = angle E, then Δ ABC = Δ DEF.

4. AAS = AAS Congruence Postulate - if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.Example: Given Δ ABC and Δ DEF,if angle A = angle D, angle C = angle F, and BC = EF , then Δ ABC = Δ DEF.

5. HL = HL Congruence Postulate - if the leg and hypotenuse of one right triangle is congruent to the corresponding leg and hypotenuse of another right triangle, then the two triangles are congruent by hypotenuse - leg postulate.Example: Given right Δ ABC and right Δ DEF, if angle B and angle E are both right angles and leg AB = leg DE and hypotenuse AC = hypotenuse DF, then Δ ABC = Δ DEF.

Friday, October 10, 2008

Geometry Chapter 3 - Sections 3.5 - 3.8 - 2008 - 2009

3.5 - 3.8 Postulates, Theorems and Proof:

I. Postulate or axiom - a true obvious statement and accepted without proof.
- is a statement whose truth is accepted without proof.

II. Theorem - is a statement that is proved by deductive reasoning.

III. Postulates:

A. Reflexive Postulate: angle A = angle A : everything is congruent to itself

B. Symmetric Postulate: if angle A = angle B then angle B = angle A (somewhat like the converse)

C. The Substitution Postulate: a quantity may be substituted for its equal in any statement of equality.

Example: if x = y and y = 8, then we can conclude by substitution that x = 8.

D. Partition Postulate: a whole is equal to the sum of the parts (part + part = whole)

E. Addition Postulate: if a = b and c = d, then a + c = b + d

F. Subtraction Postulate: if a = b and c = d, then a - c = b - d

G. Multiplication Postulate: if a = b and c = d, then ac = bd

H. Division Postulate: if a = b and c = d, then a/c = b/d.

I. Power Postulate: if a = b then a2 = b2

One of my students thought this would be a better way:

if a = b and c = d then ac = bd

Does anyone see a flaw in this?

J. Roots Postulate: if a = b and a is positive, then the square root of a = the square root of b

Example using a proof:

Monday, October 6, 2008

Chapter 3.4 Proofs

3 - 4 Direct Proofs: a proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved.

Examples we did in class:

Example1:
Example 2:
Example 3:
Example 4:
Example 5:

Thursday, October 2, 2008

Unit 3 - Proving Statements in Geometry

Vocabulary - see blue sheet
A. Definitions:

1. Midpoint of a segment - divides it into two congruent segments.
2. Bisector of a segment or an angle - divides it into two congruent parts
3. Complementary Angles - are two angles whose sum is 90 degrees.
4. Supplementary Angles - are two angles whose sum is 180 degrees.
5. Perpendicular Lines - form right angles.
6. Perimeter of a polygon - is the sum of the lengths of the sides of a polygon.
7. Scalene triangle - has no congruent sides.
8. Isosceles Triangle - has at least two congruent sides.
9. Equilateral Triangle - has three congruent sides.
10. Altitude of a triangle - is a line drawn perpendicular from a vertex to the opposite side.
11. Median of a triangle - is a line drawn from a vertex to the midpoint of the opposite side and divides the opposite side into two congruent parts.
12. Parallelogram - is a quadrilateral with both pairs of opposite sides parallel.

B. Postulates:
1. Reflexive Postulate
- any quantity is congruent to itself. angle A @ angle A
2. Transitivity Postulate - If angle A @ angle B, and angle B @ angle C, then angle A @ angle C.
3. Substitution Postulate - If angle A @ angle B, and angle A + angle C = 150 degrees, then by substitution, angle B + angle C = 150 degrees.
4. Partition Postulate - the part + the part = the whole. AB + BC = AC
5. Addition Postulate - If congruent quantities are added to congruent quantities, then their sums are congruent.
angle A @ angle B and angle C @ angle C, (angle C does not equal zero degrees)
then angle A + angle C = angle B + angle C.
6. Subtraction Postulate - If congruent quantities are subtracted from congruent quantities, then their differences are congruent.
angle A @ angle B and angle C @ angle C, (angle C does not equal zero degrees)
then Angle A - angle C = angle B - angle C.
7. Doubles Postulate - if two quantites are equal, then double their quantities are equal.
angle A = angle B, then 2(angle A) = 2(angle B).
8. Halves Postulate - if two quantities are equal, then half their quantities are equal.
angle A = angle B, then (1/2)(angle A) = (1/2)(angle B).

C. Theorems:
1. All right angles are congruent.
2. If two angles form a linear pair, then they are supplementary.
3. If two angles are supplements of the same angle, then they are congruent to each other.
3b. If two angles are complements of the same angle, then they are congruent to each other.
4. If two angles are congruent, then their supplements are congruent.
4b. If two angles are congruent, then their complements are congruent.
5. Vertical angles are congruent.
6. Corresponding parts of congruent triangles are congruent (CPCTC).
7. If two sides of a triangle are congruent, then the angles opposite are congruent.
7b. If two angles of a triangle are congruent, then the sides opposite are congruent.
8. An equilateral triangle is equiangular.
8b. An euqiangular triangle is equilateral.
9. If two lines form congruent adjacent angles, then the lines are perpendicular.
10 The supplement fo a right angle is a right angle.
11. If two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel.
11b. If two lines are cut by a transversal forming congruent corresponding angles, then the lines are parallel.
11c. If two lines are cut by a transversal forming supplementary interior angles on the same side of the transversal, then then the lines are parallel.
11d. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
11e. If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
11f. If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary.
12. If two lines are perpendicular to the same line, then they are parallel.
13. If two lines are parallel to the same line, then they are also parallel.
14. Extensions and segments of parallel lines are parallel.
15. The sum of the measures of the interior angles of a triangle is 180 degrees.
16. The sum of the measures of the interior angles of a quadrilateral is 360 degrees.
17. The measure of the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.
18. The sum of the measures of the interior angles of a polygon of n sides is 180(n - 2)
19. The measure of each interior angle of a regular polygon of n sides is (180(n-2))/n
20. The sum of the measures of the exterior angles of any polygon is 360 degrees.
21. The measure of each exterior angle of a regular polygon of n sides is 360/n.

3 - 1 Inductive Reasoning - uses a series of particular examples to lead to a general conclusion.

A. Inductive Reasoning is a powerful tool in discovering and making Conjectures - definition= generalizations arising from direct measurements of specific cases.

B. Care must be taken when applying inductive reasoning to ensure that all revelant examples are examined (no counterexample exists).

C. Inductive reasoning does not prove or explain the conjectures.

Based on these three triangles, what conjecture can you make about isosceles triangles?

Answer: If 2 sides of a triangle are congruent, then the opposite angles are congruent.

Example 2: Equilateral triangles - draw the 3 midpoints of the sides and connect them.
Make a conjecture about the 4 triangles that are formed.

Answer: It seems that it forms 4 congruent triangles

Example 3: If we draw parallelograms with opposite sides congruent by using both sides of the ruler, what conjectures can we form?

A. Opposite sides of a parallelogram are congruent.
B. It seems that all parallelograms have two acute angles and two obtuse angles.

The second conjecture is not true, if we make a parallelogram that is a rectangle, you can see all four angles are right angles, thus disproving conjecture B. This is called a counterexample - an example that shows the general conclusion is false.

You can see with 6 points, it should be 26 - 1 = 25 = 32 areas.

My students said that each time we added a point on the circle, it doubled the areas. They found out this is not true because they only had 30 or 31 areas.

In the left circle, there are 31 areas and in the right circle, there are only 30 areas. So this conjecture does not work.

This is why you have to be careful using inductive reasoning.

3 - 2 Deductive Reasoning
A. Deductive Reasoning - uses the laws of logic to combine definitions and general statements that we know to be true to reach a valid conclusion.

Every Good definition can be written as a true biconditional:

hypothesis if and only if conclusion

example:
Conditional statement: If a triangle has 2 congruent sides, then the angles opposite are congruent.

Converse: If a triangle has 2 congruent angles, then the sides opposite are congruent.

Since both the conditional and converse statements are true, then we can write the definition in biconditional form:

A triangle has 2 congruent sides if and only if the angles opposite are congruent.

Lets try another example:
Statement: A right triangle is a triangle with one right angle.

Conditional: If a triangle is a right triangle, then it has one right angle.
Converse: If a triangle has one right angle, then it is a right triangle.

Biconditional: A triangle is a right triangle if and only if it has one right angle.

3 - 3 Deductive Reasoning
1. A Proof in geometry is a valid argument that establishes the truth of a statement.
1. we use definitions, postulates and already proven theorems to show the truth of the statement.
Example: Recall the exercise with the equilateral triangle, we believed that it formed 4 congruent equilateral triangles. Using deductive reasoning, we can prove this conjecture to be true.

If 2 sides of a triangle are congruent, then the angles opposite are congruent. Each angle in an equilateral triangle measures 60 degrees so angle A = 60 degrees. That leaves angle AFB and angle BFA sum to be 180 - 60 = 120 degrees, (the sum of the angles in a triangle is 180 degrees). Since they are equal in measure, they each measure 60 degrees, forming a triangle with 3 angles measuring 60 degrees so this means it is an equiangular triangle (by definition of equiangular triangles) and equiangular triangles are equilateral triangles, so therefore each of the four triangles would have all three sides congruent (definition of equilateral triangle) forming 4 congruent equilateral triangles (all sides and angles are congruent).

How do we write a proof?

1. Two-Column Proof: has numbered statements and reasons that show the logivcal order of an argument.
2. Paragraph proof: a proof can be written in paragraph form. What we just did was a form of paragraph proof. You have to write the statements and give the reasons as part of your paragraph.
3. Flow proof: a chart that has arrows going from one statement to the next with the reasons written underneath the statement.

We will mainly use two-column proof:

Here are a few examples of algebraic proofs:

Just given some statement, we can conclude from the statement a new statement and have a reason for the statement.

Example:
Given: angle one and angle two are complementary, what can we conclude?
Conclusion: that the sum of angle one and angle two = 90 degrees by definition of complementary angles

Given: Ray BD bisects angle ABC
Conclusion: angle ABD is congruent to angle BDC by definition of angle bisector

Given: Line AB bisects line segment DE at point F
Conclusion: line segment DF is congruent to line segment FE by definition of segment bisector

Given: 0 is less than the measure of angle A which is less than 90 degrees.
Conclusion: angle A is an acute angle by definition of acute angle.

Given: B is between point A and C on line AC.
Conclusion: AB + BC = AC by partition postulate.

As you can see, you have been doing proofs, just not formally for awhile. We will continue these notes as we go along in our learning of proof writing. This is for a basic class in High School. There are many more ways to write proofs along with many more vocabulary words.