Chapter 3 Lines in a Plane

Line Relationships Intersecting lines – always coplanar Perpendicular Oblique (not ┴) Non-intersecting lines – maybe coplanar Parallel (coplanar) Skew (non-coplanar)

Theorem. - Transitivity of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other. l 3 l 3 l 1 l 2 l 1 l 2 l 2 || l 3 l 1 || l 2 l 1 || l 3

Theorem – Property of Perpendicular Lines If two coplanar lines are perpendicular to the same line, then they are parallel to each other. l 1 l 2 l 3 l 1 ┴ l 2 l 2 || l 3 l 1 ┴ l 3

Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. This an enabling postulate, allows us to draw the parallel line. Any other coplanar line through the point will eventually intersect the line.

Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. This an enabling postulate, allows us to draw the perpendicular line. Any other coplanar line through the point will not be perpendicular to the line.

Coordinate Geometry – Solutions to systems y = m1x + b1 y = m2x + b2 line 1 line 2 m1 = m2, b1 ≠ b2 m1 ≠ m2 m1 = m2, b1 = b2 no intersection no solution one point intersection unique solution coincident (same) lines infinitely many solutions

Using Laws of Logic Negation Contrapositive Structure of an argument Law of syllogism Law of detachment

Review conditional statements If a bird is a pelican, then it eats fish. p  q hypothesis conclusion Converse If a bird eats fish, then it is a pelican q  p Bi-conditional Two angles are complementary iff the sum of their measures is 90. p  q Negation – formed by denying the hypothesis and conclusion If a bird is not a pelican, then it does not eat fish. ~ p  ~ q negation doesn’t tell us what it does eat, simply that it is not fish

Contrapositive – formed by writing the negation of the converse of a statement If a bird is not a pelican, then it does not eat fish. ~ p  ~ q If a bird does not eat fish, then it is not a pelican. ~ q  ~ p The truth value of a conditional statement and its contrapositive are always the same. They are either both true or both false.

Practice p  q q  p conditional If a fruit is a banana, then it is high in potassium. write each of the following negation ~ p  ~ q If a fruit is not a banana, then it is not high in potassium. converse q  p If a fruit is high in potassium, then it is a banana. contrapositive ~ q  ~ p If a fruit is not high in potassium, then it is not a banana.

Structure of a Logical Argument 1. Theorem – Hypothesis, Conclusion 2. Argument body – Series of logical statements, beginning with the Hypothesis and ending with the Conclusion. 3. Restatement of the Theorem. (I told you so)

If you are careless with fire, Then a fish will die. If you are careless with fire, Then there will be a forest fire. If there is a forest fire, Then there will be nothing to trap the rain. If there is nothing to trap the rain, Then the mud will run into the river. If the mud runs into the river, Then the gills of the fish will get clogged with silt If the gills of the fish get clogged with silt, Then the fish can’t breathe. If a fish can’t breath, Then a fish will die If you are careless with fire, Then a fish will die.

If Hans goes to Insbruck, then the town will be smothered If Hans goes to Insbruck, then he will go skiing. If Hans begins to yodel, then he will start an avalanche. If Hans is inspired, then he will begin to yodel. If he goes skiing, then he will be inspired. If there is an avalanche, then the town will be smothered.

If Hans goes to Insbruck, then the town will be smothered If Hans goes to Insbruck, then he will go skiing. If he goes skiing, then he will be inspired. If Hans is inspired, then he will begin to yodel. If Hans begins to yodel, then he will start an avalanche. If there is an avalanche, then the town will be smothered.

Laws of logic Law of syllogism p  q q  r ∴ p  r domino theory is true Law of detachment p  q p is true ∴ q is true must tip first domino

If Mike visits Norfolk, then he will go to Busch Gardens. example: assume each statement is true If Mike visits Norfolk, then he will go to Busch Gardens. If Mike goes to Busch Gardens, then he will ride the Drachen Fire. Can you conclude Mike rode the Drachen Fire? p: Mike visits Norfolk q: Mike goes to Busch Gardens r: Mike rides the Drachen Fire. Since p  q, & q  r, are true law of syllogism infers p  r. Did Mike ride the Drachen Fire? Need law of detachment. Mike went to Norfolk last July. ∴ Mike rode the Drachen Fire.

If the dresser drawer is opened, then a mouse will jump out example: If the cat does not run out of the room, then Chad will not scream. If Chad screams then the cat runs out of the room contrapositive If a mouse jumps out of the dresser, then Chad will scream If the dresser drawer is opened, then a mouse will jump out Chad opened the dresser drawer If the dresser drawer is opened, then a mouse will jump out If a mouse jumps out of the dresser, then Chad will scream If Chad screams ? , then the cat runs out of the room The cat runs out of the room 123/1 – 5, 7 – 13 odd, 15, 20 – 24

Lesson 3.4 – No Name Theorems Th. 3.3 - If two lines are perpendicular, then they intersect to form 4 right angles. Def. of perpendicular lines – If two lines meet to form a right angle then they are perpendicular. l 1 4 1 l 2 3 2 statement reason 2. l 1 ┴ l 2 3. 1, 2, 3, & 4 are right angles 3. ┴ lines meet & form 4 rt. ’s

Lesson 3.4 – No Name Theorems Th. 3.4 – All right angles are congruent. use flow proof If two angles are both right angles, then they are congruent Given: 1 & 2 are right angles Prove: 1  2 2 1 1 is a rt.  m 1 = 90 Given Def. rt. angle talk about destination and ways we know the angles are congruent. m 1 = m 2 1  2 Transitive prop = Def  angles 2 is a rt.  m 2 = 90 Given Def. rt. angle

Lesson 3.4 – No Name Theorems Th. 3.4 – All right angles are congruent. 2 column proof If two angles are both right angles, then they are congruent Given: 1 & 2 are right angles Prove: 1  2 2 1 statement reason 1. 1 is a rt.  1. Given 2. 2 is a rt.  2. Given talk about destination and ways we know the angles are congruent. 3. m 1 = 90 3. Def. rt. angle 4. m 2 = 90 4. Def. rt. angle 5. m 1 = m 2 5. Transitive prop = 6. 1  2 6. Def  angles

Lesson 3.4 – No Name Theorems Th. 3.5 – If two lines intersect to form a pair of adjacent congruent angles, then the lines are perpendicular. l 1 Given: 1   2 1 2 l 2 Prove: l 1  l 2 statement reason 1. 1   2 1. Given 2. 1 & 2 are a linear pair 2. Def. of a linear pair 3. m 1 + m 2 = 180 3. Linear pair postulate 4. m 1 = m 2 4. Definition of congruent angles 5. m 1 + m 1 = 180 5. Substitution 6. 2(m 1)= 180 6. Distributive property 7. m 1 = 90 7. Division prop = 8. 1 is a right angle 8. Def. of a right angle 9. l 1  l 2 9. Def. of perpendicular lines

Assignment: 129/9, 10 133/1 – 4, 7 – 11 write a two column proof for both problems 10 & 11.

10. Given: 5  6 5 7 6 Prove: 5  7 reason statement

11. Given: mA + mB = 90 mC + mB = 90 no diagram Prove: A  C reason statement

Angles formed by a Transversal A transversal is a line that intersects two or more coplanar lines at different points exterior 1 1 &  3, linear pair 2 5 &  8 vertical angles 3 4 1 &  5 corresponding angles 6 &  3 alternate interior angles 2 &  7 alternate exterior angles interior 4 &  6 consecutive interior angles 6 5 exterior 8 7 alternating – opposite sides of transversal consecutive – same side of transversal

Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 m Given: m || n Conclusion: 1   2 n 2 statement reason 1. m || n 2. 1   2 2. Corresponding Angles Post.

Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. m 1 Given: m || n Conclusion: 1   2 n 2 3 create a loop on diagram, use transitive property statement reason 1. m || n 1. Given 2. 1   3 2. Corresponding Angles Post. 3. 2   3 3. Vertical Angles Theorem 4. 1   2 4. Transitive Property

Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. m 3 1 Given: m || n Conclusion: 1 &  2 are supplementary n 2 statement reason 1. m || n 1. Given 2. m1 + m 3 = 180 2. Linear Pair Post. 3. 2   3 3. Corresponding Angles Post 4. m2 = m 3 4. Def. of Congruent Angles 5. m1 + m 2 = 180 5. Substitution 6. 1 &  2 are supplementary 6. Def of Supplementary Angles

Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. m 1 Given: m || n Conclusion: 1   2 3 n 2 create a loop on diagram, use transitive property statement reason 1. m || n 1. Given 2. 1   3 2. Corresponding Angles Post. 3. 2   3 3. Vertical Angles Theorem 4. 1   2 4. Transitive Property

Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the second line. r 1 m Given: r ⊥ m m || n Conclusion: r ⊥ n 2 n statement reason 1. r ⊥ m, m || n 1. Given 2. 1 is a right angle 2. Def. of Perpendicular Lines 3. m 1 = 90 3. Def of Right Angle 4. 1   2 4. Corresponding Angles Post. 5. m 1 = m 2 5. Def. of Congruent Angles 6. m 2 = 90 6. Substitution 7. 2 is a right angle 7. Def of Right Angle 8. r ⊥ n 8. Def. of Perpendicular Lines

Proving Lines are Parallel Corresponding Angles Converse Alternate Interior Angles Converse Consecutive Interior Angles Converse Alternate Exterior Angles Converse

Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 1 m Given: 1  2 n 2 Conclusion: m || n statement reason 1. 1   2 2. m || n 2. Corresponding Angles Converse

Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. m 1 Given: 1  2 n 2 Conclusion: m || n statement reason 1. 1   2 2. m || n 2. Alternate Interior Angles Converse

Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. m 3 Given: 1 &  2 are supplementary 1 Conclusion: m || n n 2 statement reason 1. 1 &  2 are supplementary 2. m || n 2. Consecutive Interior Angles Converse

Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. m 1 Given: 1  2 n Conclusion: m || n 2 statement reason 1. 1   2 2. m || n 2. Alternate Exterior Angles Converse

1 m What do we know about this diagram? n 2 144/1 – 16 Test Moved to Wednesday 11/2

Circular Reasoning 1 m What do we know about this diagram? n 2 Parallel Lines? Circular Reasoning Law of Detachment!! Congruent Angles?

Circular Reasoning 1 m What do we know about this diagram? n 2 Parallel Lines? Circular Reasoning Law of Detachment!! Congruent Angles?