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Chapter 3
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3.1 Lines and Angles
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First thing we’re going to do is travel to another dimension
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THE THIRD DIMENSION First thing we’re going to do is travel to another dimension
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Once we get there we’ll discuss Parallel, Perpendicular, and Skew lines First thing we’re going to do is travel to another dimension
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Diagramed
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Parallel lines Diagramed
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Parallel lines Coplanar lines that don’t intersect.
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Parallel lines AB, CF, EG, DJ Diagramed
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Parallel lines AB, CF, EG, DJ AD, BJ, FG, CE Diagramed
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Perpendicular lines Diagramed
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Perpendicular lines Intersect to make a right angle
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Perpendicular lines AB and BJ Intersect to make a right angle
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Perpendicular lines AB and BJ AB and BC BJ and BC Intersect to make a right angle
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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?
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Skew lines Something perhaps that’s “gnu”
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Skew lines Something perhaps that’s “gnu”
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Skew lines Defines as lines in different planes that are not parallel.
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Skew lines The only reason they don’t intersect is because they are not coplanar.
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Skew lines Examples:
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Skew lines AB and EJ Examples:
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Skew lines AB and EJ JD and FG Examples:
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Skew lines AB and EJ JD and FG DG and CE Examples:
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Make sure that…
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Given a diagram: Make sure that…
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Given a diagram: -Identify the relationship between a pair of lines Make sure that…
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Given a diagram: -Identify the relationship between a pair of lines. -Label lines so that the desired relationship is shown Make sure that…
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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
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Use above it as a guide if you desire. Complete the Got It? on page 141
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Returning to the flat world…
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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…
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Their description and location is based upon a transversal. Returning to the flat world…
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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…
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These will break down into interior and exterior locations.
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Page 141 in your book. These will break down into interior and exterior locations.
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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.
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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.
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Now we throw in alternate which involves opposite sides of the transversal.
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Page 142 Now we throw in alternate which involves opposite sides of the transversal.
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Names and Descriptions The “glowing” line is the transversal.
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Names and Descriptions Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
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3 and 6 Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
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4 and 5 Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.
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Names and Descriptions Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
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1 and 8 Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
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2 and 7 Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.
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Names and Descriptions Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
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3 and 5 Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
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4 and 6 Same-side interior angles are nonadjacent angles that line on the same side of the transversal.
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Names and Descriptions Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
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1 and 5 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
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3 and 7 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
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2 and 6 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
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4 and 8 Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.
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Homework Page 144 – 145 11 – 24, 30 – 35, 37 – 42 Answer the questions, identify the desired relationships.
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3.2 – 3.3
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Here’s what you’re going to do…
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1)On a sheet of notebook, darken in 2 horizontal lines a few inches apart. Here’s what you’re going to do…
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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…
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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…
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Now for the thought process:
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What is special about the lines you darkened? Now for the thought process:
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What is special about the lines you darkened? They are parallel Now for the thought process:
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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:
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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:
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This is not a coincidence
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If a transversal intersects 2 parallel lines: This is not a coincidence
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If a transversal intersects 2 parallel lines: (1)Alternate interior angles are congruent. This is not a coincidence
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If a transversal intersects 2 parallel lines: (1)Alternate interior angles are congruent. (2)Alternate exterior angles are congruent. This is not a coincidence
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If a transversal intersects 2 parallel lines: (2)Alternate exterior angles are congruent. (3)Corresponding angles are congruent. This is not a coincidence
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If a transversal intersects 2 parallel lines: (3)Corresponding angles are congruent. (4)Same side interior angles are supplementary. This is not a coincidence
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A postulate…
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3.1 If a transversal intersects two parallel lines, then same side interior angles are supplementary A postulate…
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3.1 If a transversal intersects two parallel lines, then same side interior angles are supplementary A list of theorems
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3.1 If a transversal intersects two parallel lines, then alternate interior angles are congruent A list of theorems
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3.2 If a transversal intersects two parallel lines, then corresponding angles are congruent A list of theorems
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3.3 If a transversal intersects two parallel lines, then alternate exterior angles are congruent. A list of theorems
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The long way to find angle measures. 12 3 4 56 7 8 Let m 3 = 82
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The long way to find angle measures. 12 3 4 56 7 8 Let m 3 = 82 -m 2 = ____ -m 1 = ____ -m 4 = ____
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Vertical angle conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 2 = 82 -m 1 = ____ -m 4 = ____
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Linear Pair Angle Conjecutre 12 3 4 56 7 8 Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = ____
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Linear Pair or Vertical Angle Conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = 82
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Now we march on to the other point of intersection 12 3 4 56 7 8 Let m 3 = 82 -m 2 = 82 -m 1 = 98 -m 4 = 82
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Now we march on to the other point of intersection 12 3 4 56 7 8 Let m 3 = 82 -m 5 = ____ -m 6 = ____ -m 7 =____ -m 8 =____
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By the Same-Side Conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 5 = ____ -m 6 = ____ -m 7 =____ -m 8 =____
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By the Same-Side Conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = ____ -m 7 =____ -m 8 =____
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By the Linear Pair Conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____
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By the Linear Pair Conjecture 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____ What is the defined relationship between 3 and 6?
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Alternate Interior Angles!!! 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =____ -m 8 =____ What is the defined relationship between 3 and 6?
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Alternate Interior Angles!!! 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 =82 -m 8 =____
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Which has a corresponding angle relationship with 3. 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 =____
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Which has a corresponding angle relationship with 3. 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 = 98
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Which makes alternate exterior angle magic with 1. 12 3 4 56 7 8 Let m 3 = 82 -m 5 = 98 -m 6 = 82 -m 7 = 82 -m 8 = 98
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Now the short method…
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If you’re asked to find, not justify or prove that angles are congruent or have the same angle measure:
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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 .
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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
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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
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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
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Formal proof Given: j || k Prove: 4 6
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Formal proof StatementReason
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Formal proof StatementReason m 3 + m 4 = 180
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Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture
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Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture m 3 + m 6 = 180
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Formal proof StatementReason m 3 + m 4 = 180Linear Pair Conjecture m 3 + m 6 = 180Same Side Interior Angle Conjecture
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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
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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
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Formal proof StatementReason m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality 4 6
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Formal proof StatementReason m 3 + m 4 = m 3 + m 6 Transitive Property m 4 = m 6Subtraction Property of Equality 4 6Definition of Congruence
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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
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If it does ask you to justify…
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Include a definition or theorem that allows you to state your angle relationship.
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If it does ask you to justify… Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1
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Why is 3 also 132? Include a definition or theorem that allows you to state your angle relationship. 3.2 Practice # 1
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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
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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:
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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
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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:
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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
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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
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Solution 5 is 78 because of alternate interior angles.
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Solution 5 is 78 because of alternate interior angles. 1 is 78 because of vertical angles.
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Be specific!!! 1 is 78 because of vertical angles. 7 is 78 because of corresponding angles
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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
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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
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Similar idea, moving to #5
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130 is the reference angle. Similar idea, moving to #5
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130 is the reference angle. Angle 1 is _____ because it makes a __________ ________ with the 130 angle. Similar idea, moving to #5
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130 is the reference angle. Angle 1 is 50 because it makes a linear pair with the 130 angle. Similar idea, moving to #5
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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
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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
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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
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Things to remember in sketches:
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Make sure their exists a relationship between the angles. Things to remember in sketches:
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Make sure their exists a relationship between the angles. Touch the same transversal, and that the lines are parallel. Things to remember in sketches:
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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:
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Reversing the process
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It seems we’ve gone down this road before…
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Theorem 3-4
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If 2 lines and a transversal form corresponding angles that are congruent, then the lines are parallel
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Theorem 3-5 If 2 lines and a transversal form alternate interior angles that are congruent, then the lines are parallel
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Theorem 3-6 If 2 lines and a transversal form same side interior angles that are supplementary, then the lines are parallel
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Theorem 3-7 If 2 lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel
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1 – 6 Let’s use the 3.3 Practice to see how to problem solve…
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1 – 6 (A)Find the congruent angles Let’s use the 3.3 Practice to see how to problem solve…
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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…
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1 – 6 (A)Find the congruent angles (B)Determine the lines they are on. NOT THE TRANSVERSAL!!!
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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…
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7:Poof… A proof… Let’s use the 3.3 Practice to see how to problem solve…
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8: A walk through… Let’s use the 3.3 Practice to see how to problem solve…
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9 – 14 Let’s use the 3.3 Practice to see how to problem solve…
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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…
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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…
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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.
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15 – 20: Some of my faves…
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(1)Look at the angle pair they provide you. 15 – 20: Some of my faves…
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(1)Look at the angle pair they provide you. (2)Identify the relationship, if any, from the diagram. 15 – 20: Some of my faves…
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(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…
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#RelationshipJustification 15 – 20: Some of my faves…
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#RelationshipJustification 15 11 & 10 are supplementary 15 – 20: Some of my faves…
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#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…
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#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…
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#RelationshipJustification 16 6 9Lines a and b are parallel because alternate interior angles are congruent. 15 – 20: Some of my faves…
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#RelationshipJustification 16 6 9Lines a and b are parallel because alternate interior angles are congruent. You fill in the rest…
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#RelationshipJustification 1713 and 14 supplementary Nothing: this is always true no matter what lines are parallel. 1813 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…
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#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…
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