Chapter 3
3.1 Lines and Angles
First thing we’re going to do is travel to another dimension
THE THIRD DIMENSION First thing we’re going to do is travel to another dimension
Once we get there we’ll discuss Parallel, Perpendicular, and Skew lines First thing we’re going to do is travel to another dimension
Diagramed
Parallel lines Diagramed
Parallel lines Coplanar lines that don’t intersect.
Parallel lines AB, CF, EG, DJ Diagramed
Parallel lines AB, CF, EG, DJ AD, BJ, FG, CE Diagramed
Perpendicular lines Diagramed
Perpendicular lines Intersect to make a right angle
Perpendicular lines AB and BJ Intersect to make a right angle
Perpendicular lines AB and BJ AB and BC BJ and BC Intersect to make a right angle
Perpendicular lines AB and BJ AB and BC BJ and BC If 2 lines are perpendicular to the same line, are they perpendicular to each other?
Skew lines Something perhaps that’s “gnu”
Skew lines Something perhaps that’s “gnu”
Skew lines Defines as lines in different planes that are not parallel.
Skew lines The only reason they don’t intersect is because they are not coplanar.
Skew lines Examples:
Skew lines AB and EJ Examples:
Skew lines AB and EJ JD and FG Examples:
Skew lines AB and EJ JD and FG DG and CE Examples:
Make sure that…
Given a diagram: Make sure that…
Given a diagram: -Identify the relationship between a pair of lines Make sure that…
Given a diagram: -Identify the relationship between a pair of lines. -Label lines so that the desired relationship is shown Make sure that…
Given a diagram: -Identify the relationship between a pair of lines. -Label lines so that the desired relationship is shown Complete the Got It? on page 141
Use above it as a guide if you desire. Complete the Got It? on page 141
Returning to the flat world…
On a plane, when lines intersect two or more lines at distinct points, the angles formed at these points create special angle pairs. Returning to the flat world…
Their description and location is based upon a transversal. Returning to the flat world…
Their description and location is based upon a transversal. A line that intersects two or more lines at distinct points. Returning to the flat world…
These will break down into interior and exterior locations.
Page 141 in your book. These will break down into interior and exterior locations.
Page 141 in your book. -Interior angles are found between the 2 lines that are intersected These will break down into interior and exterior locations.
Page 141 in your book. -Interior angles are found between the 2 lines that are intersected -As you can guess, exterior angles are then found outside these same lines. These will break down into interior and exterior locations.
Now we throw in alternate which involves opposite sides of the transversal.
Page 142 Now we throw in alternate which involves opposite sides of the transversal.
Names and Descriptions The “glowing” line is the transversal.
Names and Descriptions Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
3 and 6 Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
4 and 5 Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
Names and Descriptions Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
1 and 8 Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
2 and 7 Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
Names and Descriptions Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
3 and 5 Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
4 and 6 Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
Names and Descriptions Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
1 and 5 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
3 and 7 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
2 and 6 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
4 and 8 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
Homework Page 144 – – 24, 30 – 35, 37 – 42 Answer the questions, identify the desired relationships.
3.2 – 3.3
Here’s what you’re going to do…
1)On a sheet of notebook, darken in 2 horizontal lines a few inches apart. Here’s what you’re going to do…
1)On a sheet of notebook, darken in 2 horizontal lines a few inches apart. 2)Create a transversal that is not perpendicular to your 2 lines. Here’s what you’re going to do…
2)Create a transversal that is not perpendicular to your 2 lines. 3)Measure all 8 angles that are formed by the trans- versal and the lines you darkened. Here’s what you’re going to do…
Now for the thought process:
What is special about the lines you darkened? Now for the thought process:
What is special about the lines you darkened? They are parallel Now for the thought process:
What is special about the lines you darkened? They are parallel What is special about pairs of angles you measured? Now for the thought process:
What is special about the lines you darkened? They are parallel What is special about pairs of angles you measured? They are congruent Now for the thought process:
This is not a coincidence
If a transversal intersects 2 parallel lines: This is not a coincidence
If a transversal intersects 2 parallel lines: (1)Alternate interior angles are congruent. This is not a coincidence
If a transversal intersects 2 parallel lines: (1)Alternate interior angles are congruent. (2)Alternate exterior angles are congruent. This is not a coincidence
If a transversal intersects 2 parallel lines: (2)Alternate exterior angles are congruent. (3)Corresponding angles are congruent. This is not a coincidence
If a transversal intersects 2 parallel lines: (3)Corresponding angles are congruent. (4)Same side interior angles are supplementary. This is not a coincidence
A postulate…
3.1 If a transversal intersects two parallel lines, then same side interior angles are supplementary A postulate…
3.1 If a transversal intersects two parallel lines, then same side interior angles are supplementary A list of theorems
3.1 If a transversal intersects two parallel lines, then alternate interior angles are congruent A list of theorems
3.2 If a transversal intersects two parallel lines, then corresponding angles are congruent A list of theorems
3.3 If a transversal intersects two parallel lines, then alternate exterior angles are congruent. A list of theorems
The long way to find angle measures Let m 3 = 82
The long way to find angle measures Let m 3 = 82 -m 2 = ____ -m 1 = ____ -m 4 = ____
Vertical angle conjecture Let m 3 = 82 -m 2 = 82 -m 1 = ____ -m 4 = ____
Linear Pair Angle Conjecutre Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = ____
Linear Pair or Vertical Angle Conjecture Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = 82
Now we march on to the other point of intersection Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = 82
Now we march on to the other point of intersection Let m 3 = 82 -m 5 = ____ -m 6 = ____ -m 7 =____ -m 8 =____
By the Same-Side Conjecture Let m 3 = 82 -m 5 = ____ -m 6 = ____ -m 7 =____ -m 8 =____
By the Same-Side Conjecture Let m 3 = 82 -m 5 = 98 -m 6 = ____ -m 7 =____ -m 8 =____
By the Linear Pair Conjecture Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____
By the Linear Pair Conjecture Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____ What is the defined relationship between 3 and 6?
Alternate Interior Angles!!! Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____ What is the defined relationship between 3 and 6?
Alternate Interior Angles!!! Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =82 -m 8 =____
Which has a corresponding angle relationship with Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 =____
Which has a corresponding angle relationship with Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 = 98
Which makes alternate exterior angle magic with Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 = 98
Now the short method…
If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure:
Now the short method… If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure: - All acute angles are .
Now the short method… If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure: - All acute angles are . - All obtuse angles are
Now the short method… If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure: - All acute angles are . - All obtuse angles are - The sum of an acute and an obtuse angle = 180
Provided the lines are parallel. If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure: - All acute angles are . - All obtuse angles are - The sum of an acute and an obtuse angle = 180
Formal proof Given: j || k Prove: 4 6
Formal proof StatementReason
Formal proof StatementReason m 3 + m 4 = 180
Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture
Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture m 3 + m 6 = 180
Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture m 3 + m 6 = 180Same Side Interior Angle Conjecture
Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture m 3 + m 6 = 180Same Side Interior Angle Conjecture m 3 + m 4 = m 3 + m 6 Transitive Property
Formal proof StatementReason m 3 + m 6 = 180Same Side Interior Angle Conjecture m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality
Formal proof StatementReason m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality 4 6
Formal proof StatementReason m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality 4 6Definition of Congruence
You will most likely have to do one of these on your next quiz. StatementReason m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality 4 6Definition of Congruence
If it does ask you to justify…
Include a definition or theorem that allows you to state your angle relationship.
If it does ask you to justify… Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles 5:
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles 5: Corresponding
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles 5: Corresponding 7:
Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles 5: Corresponding 7:Alternate Exterior
You try #2 Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1 3:Vertical Angles 5: Corresponding 7:Alternate Exterior
Solution 5 is 78 because of alternate interior angles.
Solution 5 is 78 because of alternate interior angles. 1 is 78 because of vertical angles.
Be specific!!! 1 is 78 because of vertical angles. 7 is 78 because of corresponding angles
Be specific!!! 1 is 78 because of vertical angles. 7 is 78 because of corresponding angles Alternative: 7 makes a vertical angle pair with #5
If you don’t write anything, we assume you are talking about the angle measure given to you. 1 is 78 because of vertical angles. 7 is 78 because of corresponding angles Alternative: 7 makes a vertical angle pair with #5
Similar idea, moving to #5
130 is the reference angle. Similar idea, moving to #5
130 is the reference angle. Angle 1 is _____ because it makes a __________ ________ with the 130 angle. Similar idea, moving to #5
130 is the reference angle. Angle 1 is 50 because it makes a linear pair with the 130 angle. Similar idea, moving to #5
130 is the reference angle. Angle 1 is 50 because it makes a linear pair with the 130 angle. Angle 2 is Similar idea, moving to #5
130 is the reference angle. Angle 1 is 50 because it makes a linear pair with the 130 angle. Angle 2 is 130 because it is a corresponding angle to the 130. Similar idea, moving to #5
130 is the reference angle. Angle 1 is 50 because it makes a linear pair with the 130 angle. Angle 2 is 130 because it is a corresponding angle to the 130. You do #6
Things to remember in sketches:
Make sure their exists a relationship between the angles. Things to remember in sketches:
Make sure their exists a relationship between the angles. Touch the same transversal, and that the lines are parallel. Things to remember in sketches:
Make sure their exists a relationship between the angles. Touch the same transversal, and that the lines are parallel. Keep this in mind as we tackle the remaining problems. Things to remember in sketches:
Reversing the process
It seems we’ve gone down this road before…
Theorem 3-4
If 2 lines and a transversal form corresponding angles that are congruent, then the lines are parallel
Theorem 3-5 If 2 lines and a transversal form alternate interior angles that are congruent, then the lines are parallel
Theorem 3-6 If 2 lines and a transversal form same side interior angles that are supplementary, then the lines are parallel
Theorem 3-7 If 2 lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel
1 – 6 Let’s use the 3.3 Practice to see how to problem solve…
1 – 6 (A)Find the congruent angles Let’s use the 3.3 Practice to see how to problem solve…
1 – 6 (A)Find the congruent angles (B)Determine the lines they are on. Let’s use the 3.3 Practice to see how to problem solve…
1 – 6 (A)Find the congruent angles (B)Determine the lines they are on. NOT THE TRANSVERSAL!!!
1 – 6 (A)Find the congruent angles (B)Determine the lines they are on. (C)Identify the relationship between them to justify. Let’s use the 3.3 Practice to see how to problem solve…
7:Poof… A proof… Let’s use the 3.3 Practice to see how to problem solve…
8: A walk through… Let’s use the 3.3 Practice to see how to problem solve…
9 – 14 Let’s use the 3.3 Practice to see how to problem solve…
9 – 14 (A)Work under the belief that the lines are parallel. Let’s use the 3.3 Practice to see how to problem solve…
9 – 14 (A)Work under the belief that the lines are parallel. (B)Identify the relationship and set up an equation. Let’s use the 3.3 Practice to see how to problem solve…
9 – 14 (A)Work under the belief that the lines are parallel. (B)Identify the relationship and set up an equation. This will be either congruent or supplementary, if possible.
15 – 20: Some of my faves…
(1)Look at the angle pair they provide you. 15 – 20: Some of my faves…
(1)Look at the angle pair they provide you. (2)Identify the relationship, if any, from the diagram. 15 – 20: Some of my faves…
(1)Look at the angle pair they provide you. (2)Identify the relationship, if any, from the diagram. (3)Find the desired value. 15 – 20: Some of my faves…
#RelationshipJustification 15 – 20: Some of my faves…
#RelationshipJustification 15 11 & 10 are supplementary 15 – 20: Some of my faves…
#RelationshipJustification 15 11 & 10 are supplementary Lines u and t are parallel because same side interior angles are supplementary. 15 – 20: Some of my faves…
#RelationshipJustification 15 11 & 10 are supplementary Lines u and t are parallel because same side interior angles are supplementary. 16 6 9 15 – 20: Some of my faves…
#RelationshipJustification 16 6 9Lines a and b are parallel because alternate interior angles are congruent. 15 – 20: Some of my faves…
#RelationshipJustification 16 6 9Lines a and b are parallel because alternate interior angles are congruent. You fill in the rest…
#RelationshipJustification 1713 and 14 supplementary Nothing: this is always true no matter what lines are parallel and 15 are congruentLines t and u are parallel because corresponding angles are congruent 1912 is supplementary to 33 is also supplementary to 4 because of linear pairs. By the congruent supplements theorem, 4 and 12 are congruent, which are corresponding angles, making a and b parallel You fill in the rest…
#RelationshipJustification 1912 is supplementary to 33 is also supplementary to 4 because of linear pairs. By the congruent supplements theorem, 4 and 12 are congruent, which are corresponding angles, making a and b parallel 202 and 13 are congruenta and b are parallel since alternate exterior angles are congruent. You fill in the rest…