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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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Chapter 9 summary project

Jeffrey Lio

Period 2

12/18/03

Chapter 9 Summary Project


Chapter 9 summary project

Parallelism

Definitions:

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

L1 and L2 are skew lines.


Chapter 9 summary project

L1 and L2 are parallel lines.


Chapter 9 summary project

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

L1 is the transversal of L2 and L3.


Chapter 9 summary project

  • 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.


Chapter 9 summary project

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


Chapter 9 summary project

  • 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.


Chapter 9 summary project

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


Chapter 9 summary project

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


Chapter 9 summary project

  • 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


Chapter 9 summary project

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


Chapter 9 summary project

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.


Chapter 9 summary project

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.


Chapter 9 summary project

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 =


Chapter 9 summary project

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.


Chapter 9 summary project

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


Chapter 9 summary project

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.


Chapter 9 summary project

  • 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


Chapter 9 summary project

A convex quadrilateral:

A quadrilateral that is not convex:


Chapter 9 summary project

  • 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.


Chapter 9 summary project

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.


Chapter 9 summary project

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.


Chapter 9 summary project

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


Chapter 9 summary project

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

Conclusion: ABCD is a parallelogram.


Chapter 9 summary project

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.