Chapter 6 parallel lines
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Chapter 6 – Parallel Lines. By Grace Grimaldi and Matt O’Donnell. Lesson 1 – Line Symmetry. symmetric two points are symmetric with respect to a line iff the line is the perpendicular bisector of the line segment connecting the two points. Lesson 1 – Line Symmetry.

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Chapter 6 – Parallel Lines

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Chapter 6 parallel lines

Chapter 6 – Parallel Lines

By Grace Grimaldi and Matt O’Donnell


Lesson 1 line symmetry

Lesson 1 – Line Symmetry

  • symmetric

    • two points are symmetric with respect to a lineiff the line is the perpendicular bisector of the line segment connecting the two points


Lesson 1 line symmetry1

Lesson 1 – Line Symmetry

  • In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment


Lesson 1 line symmetry2

Lesson 1 – Line Symmetry

  • Construction

    • To construct a line perpendicular to a given line through a point


Lesson 2 proving lines parallel

Lesson 2 – Proving Lines Parallel

  • parallel

    • two lines are parallel iff they line in the same plane and do not intersect


Lesson 2 proving lines parallel1

Lesson 2 – Proving Lines Parallel

  • transversal

    • a line that intersects two or more lines in different points

l1

l2

t


Lesson 2 proving lines parallel2

Lesson 2 – Proving Lines Parallel


Lesson 2 proving lines parallel3

Lesson 2 – Proving Lines Parallel

Ways to Prove Lines Parallel

  • equal corresponding angles mean that lines are parallel

  • equal alternate interior angles mean that lines are parallel

  • supplementary same-side interior angles mean that lines are parallel


Lesson 2 proving lines parallel4

Lesson 2 – Proving Lines Parallel

  • In a plane, two lines perpendicular to a third line are parallel to each other


Lesson 3 the parallel postulate

Lesson 3 – The Parallel Postulate

  • Construction

    • To construct a line parallel to a given line through a given point


Lesson 3 parallel postulate

Lesson 3 – Parallel Postulate

  • The Parallel Postulate

    • Through a point not on a line, there is exactly one line parallel to the given line

. P

parallel

not parallel


Lesson 3 the parallel postulate1

Lesson 3 – The Parallel Postulate

  • In a plane, two lines parallel to a third line are parallel to each other.

parallel

parallel

parallel

parallel

parallel

parallel


Lesson 4 parallel lines and angles

Lesson 4 – Parallel Lines and Angles

  • parallel lines ⟷ equal corresponding angles


Lesson 4 parallel lines and angles1

Lesson 4 – Parallel Lines and Angles

  • parallel lines ⟷ equal alternate interior angles


Lesson 4 parallel lines and angles2

Lesson 4 – Parallel Lines and Angles

  • In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other

    • This is easy to determine because if one of the perpendicular lines forms a 90 degree angle with the transversal than the corresponding angle on the parallel line must also be 90 degrees making it perpendicular to the transversal


Lesson 4 parallel lines and angles3

Lesson 4 – Parallel Lines and Angles

  • parallel lines ⟷ supplementary same-side interior angles


Lesson 5 the angles of a triangle

Lesson 5 – The Angles of a Triangle

  • The Sum Angle Theorem

    • the sum of the angles of a triangle is 180°

    • the third angle of any triangle can be found by adding up the sum of the two known angles and subtracting that from 180


Lesson 5 the angles of a triangle1

Lesson 5 – The Angles of a Triangle

  • If two angles of one triangle are equal to two angles of another triangle, the third angles are equal.


Lesson 5 the angles of a triangle2

Lesson 5 – The Angles of a Triangle

  • The acute angles of a right triangle are complementary


Lesson 5 the angles of a triangle3

Lesson 5 – The Angles of a Triangle

  • Each angle of an equilateral triangle is 60°

    • 180/3 = 60


Lesson 5 the angles of a triangle4

Lesson 5 – The Angles of a Triangle

  • Exterior angle - the angle between any side of a polygon and an extended adjacent side

    • An exterior angle of a triangle is equal to the sum of the remote interior angles.


Lesson 6 aas and hl congruence

Lesson 6 – AAS and HL Congruence

  • AAS Congruence

    • If two angles and the side opposite one of them are equal to the corresponding parts of another triangle, the triangles are congruent.

    • If two of the angles are equal it follows that the third angles will be equal by corollary one of the angle sum theorem.


Lesson 6 aas and hl congruence1

Lesson 6 – AAS and HL Congruence

  • Hypotenuse-Leg (HL) Congruence

    • If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle than the triangles are congruent.


Lesson 6 aas and hl congruence2

Lesson 6 – AAS and HL Congruence

  • HL Congruence (continued)

    • If the two triangles are joined along the two equal legs, an isosceles triangle is formed and the far bottom corners are shown to be equal, which shows the triangles to be congruent.


Chapter 6 summary tips

Chapter 6 Summary/Tips

  • This chapter focused on parallel lines and the relationships with angles that parallel lines form. The majority of the Chapter 6 Test were problems that dealt with finding the measures of angles in a figure given one, two or no measures of other angles in the figure.

  • Important to Know

    • Angle Sum Theorem

    • AAS Congruence

    • HL Congruence

    • Exterior Angle Equality

    • Parallel lines and the angles they form


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