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Jeffrey Lio Period 2 12/18/03. Chapter 9 Summary Project. Parallelism. Definitions:. Skew Lines : Any two non-coplanar lines that do not intersect. L 1 and L 2 are skew lines. Parallel Lines : Any two coplanar lines that do not intersect. L 1 and L 2 are parallel lines.

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Jeffrey Lio

Period 2

12/18/03

Chapter 9 Summary Project


Parallelism

Definitions:

  • Skew Lines: Any two non-coplanar lines that do not intersect.

L1 and L2 are skew lines.


L1 and L2 are parallel lines.


  • Transversal: a line that intersects two coplanar lines in 2 different points.

L1 is the transversal of L2 and L3.


  • Alternate Interior Angles: any two angles on opposite sides of the transversal and are between the two lines that are cut by the transversal.

a and c are alternate interior angles.


a and d are interior angles on the same side of the transversal.


  • Corresponding Angles: When two lines are cut by a transversal, if d and f are alternate interior angles, and b is the vertical angle of d, then b and f are corresponding angles.

c and e are alternate interior angles and e and g are vertical angles, so c and g are corresponding angles too.


Theorems and Corollaries:

  • AIP Theorem: When two lines are cut by a transversal, and if a pair of alternate interior angles are congruent, then the lines are parallel.

Restatement: L1 and L2 are cut by transversal T. a and b are alternate interior angles and are congruent.

Conclusion: L1 and L2 are parallel.

Then

If


Given EGB CHF, prove that AB and CD are parallel.

Hypothesis: EGB CHF.

Conclusion: AB CD.

Statements

Reasons

  • EGB CHF

  • AGF EGB, DHE CHF

  • AGF DHE

  • AB CD

  • given

  • VAT

  • substitution

  • AIP


  • PCA Corollary : A pair of parallel lines cut by a transversal have congruent corresponding angles.

Restatement: L1 and L2 are parallel and are cut by transversal T. a and b are corresponding angles.

Conclusion: a and b are congruent.

If

Then

L1

L2

L1

L2

T

T


If BC=FG, CD=HG, and AC EG, prove BCD FHG.

Statements

Reasons

  • Given

  • PCA

  • SAS

  • BC=FG, CD=HG, AC EG

  • FGH BCD

  • BCD FHG


Triangles

Definitions:

  • Exterior Angle: d is an exterior angle of ABC because it lies in the exterior of a triangle and forms a linear pair with one of the angles of the triangle.

d is an exterior angle of ABC.

A

  • Remote Interior Angles: Given an exterior angle, the remote interior angles are the two angles that do not share a common side with the exterior angle.

a and b are remote interior angles of d and e.


Theorems and Corollaries:

  • In a triangle, the sum of the measures of its interior is equivalent to 180.

Restatement: Given ABC

Conclusion: A + B + C=180.


Prove that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.

Hypothesis: z is the exterior angle. x, y, and w are angles of the triangle.

Conclusion: m z=m w+m x

Statements:

Reasons:

  • z and y are supp

  • m y+m z=180

  • m w+m x+m y=180

  • 180=180

  • m y+m z=m w+m x+m y

  • m z=m w+m x

  • Supp Post

  • Def of supp

  • Given a triangle, sum of angles=180

  • Reflex ax of =

  • Trans ax of =

  • Sub ax of =


Restatement: Given ABC where D and E are midpoints of AB and AC respectively.

Conclusion: DE=BC/2 and DE BC.

DE=BC/2 and DE BC.


Given AB=FG, AC=FH, sides of a triangle is parallel to the third side and has a length equal to half that of the third side. BC=GH, and D, E, I, and J are midpoints, prove ADE FIJ.

Reasons:

Statements:

  • AB=FG, AC=FH, BC=GH, D, E, I, J are midpoints

  • DE=BC/2, IJ=GH/2

  • BC/2=GH/2

  • DE=IJ

  • ADE FIJ

  • given

  • midline theorem

  • Div ax of =

  • Substitution

  • SSS


Quadrilaterals sides of a triangle is parallel to the third side and has a length equal to half that of the third side.

Definitions:

  • Quadrilateral: A, B, C, and D are the endpoints of AB, BC, CD, and DA. Since no three of these points are collinear and the segments are contained in plane E, the union of these four segments is a quadrilateral.


  • Vertices of a Quadrilateral: The endpoints A, B, C, and D are vertices of the quadrilateral

  • Sides of a Quadrilateral: The individual segments whose union form the quadrilateral. AB, BC, CD, and DA are sides of ABCD.

  • Angles of a Quadrilateral: The angles formed by the union of two segments that share a common point. ABC, BCD, CDA, and DAB are angles of ABCD


A convex quadrilateral:

A quadrilateral that is not convex:


  • Opposite Sides: two sides of a quadrilateral that do not intersect

  • Opposite Angles: two angles of a quadrilateral that do not share a common side

  • Consecutive Sides: two sides of a quadrilateral that do intersect

  • Opposite Angles: two angles of a quadrilateral that do share a common side

In the figure above, AB and CD are opposite sides while AB and BC are consecutive sides. A and B are consecutive angles, and A and C are opposite angles.


Parallelogram: a quadrilateral made up of two pairs of parallel lines.

Since AB is parallel to DC and AD is parallel to BC, ABCD is a parallelogram.

  • Trapezoid: a quadrilateral with only one pair of parallel sides

  • Bases of a Trapezoid: the pair of parallel sides in the trapezoid

  • Median of a Trapezoid: a segment whose endpoints are the midpoints of the two opposite sides that are not parallel

Since AB is parallel to DC, ABCD is a trapezoid. FE is the median and AB and CD are the bases of the trapezoid.


Theorems and Corollaries: parallel lines.

A diagonal of a parallelogram divides it into two congruent triangles.

Restatement: Given a parallelogram, ABCD and diagonal AC.

Conclusion: ABC CDA.


Prove that the opposite angles of a parallelogram are congruent.

Hypothesis: ABCD is a parallelogram.

Conclusion: A C.

Statements:

Reasons:

  • ABCD is a parallelogram

  • DB=DB

  • AD BC, AB CD

  • ADB CBD, CDB ABD

  • ADB CBD

  • A C

  • Given

  • Reflex ax of =

  • Def of parallelogram

  • PAI

  • ASA

  • CPCTC


Restatement: Given ABCD where AB=CD and AB CD.

Conclusion: ABCD is a parallelogram.


Hypothesis: AB=CD and ABD CDB. it is a parallelogram.

Conclusion: ABCD is a parallelogram.

Statements:

Reasons:

  • AB=CD, ABD CDB

  • AB CD

  • ABCD is a parallelogram

  • Given

  • AIP

  • If two sides are parallel and congruent in a quadrilateral, it is a parallelogram.


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