Chapter 9

Paul HeinPeriod 212/12/2003

                                                                  

ParallelismKey Terms

Skew Lines: Skew lines are 2 lines that are neither parallel nor intersect: thus, they must be in different planes.

L1 and L2 are intersecting lines. L1 and L3 are parallel lines. L2 and L3 are Skew lines.

Transversal: A transversal is a line that intersects two coplanar lines.

L1 and L2 are coplanar. Thus, line T is the transversal.

Alternate Interior Angles: Alternate interior angles are formed when 2 lines are cut by a transversal. They are any two angles thatAre on opposite sides of the transversal, are formed by 2 different coplanar lines, and are on the interior of the parallel lines.

L1 and L2 are 2 coplanar lines. T is a transversal of them. A is an alternate interior angle toC, and B is an alternate interior angle to D.

Interior Angles on the same side of the transversal: These are exactly what they sound like: IF you have 2 coplanar lines that areCut by a transversal, then any two angles on the interior of the parallel lines and on the same side of the transversal fit this Description.

(fig above) L1 and L2 are coplanar lines. T is a transversal of them. A is an interiorAngle on the same side of the transversal to D, and B is an interior angle on the same side of the transversal with C.

Corresponding Angles: If you have two coplanar lines cut be a transversal, the angle vertical to one of the alternate interior Angles is a corresponding angle to the other alternate interior angle.

L1 and L2 are coplanar, with transversal T. A is vertical to B, and B is an alternate interior angle to C. therefore, A is a Corresponding angle to C.

                                                                               

Theorem #1:The AIP theorem

This theorem says that if two alternate interior angles are congruent, then the lines that make them are parallel. This is used to Prove two lines parallel in a proof. This can be proved because if A is congruent to C, and A is supplementary to B because Of the Linear Pair Theorem, so B is supplementary to C. Because of theorem 9-8, which states that if a pair of same-side Interior angles are supplementary, then the lines are parallel, L1 and L2 are parallel.

In simpler terms, if A and C are congruent, then L1 and L2 are parallel.

Proof of AIP theorem

A

B

Given: A is congruent toB

Prove: L1L2L1

L2

S R

1. A B 1. Given

2. L1L2 2. AIP theorem

The CAP theoremThe CAP theorem: The CAP theorem, short for the Corresponding Angle Parallel Theorem, States that given two lines with a transversal through them, if two corresponding angles areCongruent, then the two lines are parallel.

L1

L2

TA

B

If A is congruent to B, Then L1 is parallel to L2.

A

BL2

L1

TGiven: A is congruent to B

Prove: L1||L2

S R

1. AB 1.Given

2. L1||L2 2. CAP theorem

TrianglesKey Terms

Right Triangle: A right triangle is a triangle with one right angle (90). Because ofThis, we can conclude that the two other angles are acute, because all of the anglesIn a triangle must add up to 180 degrees. Thus, No other angle can be 90 or higher,Because that would exceed this rule of triangles. There are Many unique propertiesAbout a right triangle and its sides/angle measurements.

Acute angles

90

Hypotenuse: The hypotenuse is the side opposite of the right angle in a right triangle.It is always longer than the two other sides of the triangle. The ancient mathematician Pythagoras found out that if the lengths of the two other sides of the right Triangle were each squared and then added together, the answer would be the length of the hypotenuse squared.

Hypotenuse

A

B

C

Pythagorean Theorem: A²+B²=C²

TrianglesThe angles of a triangle theorem*

*Not real name

This theorem states that all of the angles of a triangle add up to 180. There is noWay to prove this theorem, but it is possible to prove that all of the angles of aTriangle measure up to less than 181.

S R

Given: A and B are complementary A

B C

Prove: C is right

1: A is comp. To B2: ma +mb=903: ma+mb+mc=1804: c+90=1805: c=906: C is right

1: Given2: Defn. of comp.3: Angles of a triangle thm.4: Substitution5: Subtracti0n prop. Of =6: Defn. of right angle

Acute angles of a Right triangle theoremThis theorem states: “the acute angles of a right triangle are complimentary”.This is because the angle of a triangle add up to 180. Since one of the angles is 90Degrees, that’s 90 off the 180 requirement. Thus, the other angles must add up to be90 degrees, because the sum of the angles of any given triangle must add up to be 180 degrees. Because they add up to 90, the other angles are complimentary.

A

X

X=90-a, and a=90-x.

Given: B is right, A=30

Prove: D=60A

C

BD

S R

1: Givens

2: a vert. To c

3: ac

4: mc=30

5: c comp. To d

6: md=60

1: Given

2: Defn. of Vert.

3: VAT

4: Substitution

5: acute s of a rt.Thm6: defn. of comp.

QuadrilateralsKey Terms

Quadrilateral: ok, draw 4 coplanar points (lets use p, q, r, and s), no three of them being collinear. Then Connect them in consecutive order (segments pq, qr, rs, and ps).Viola! Your very own quadrilateral. Your mom will be proud.

S

R

Q

P

Trapezoid: a trapezoid is a quadrilateral with one and only one pair of sidesThat are parallel. The parallel sides are called the base sides, and theNonparallel sides are called the medians.

Base sidesmedians

Parallelogram: a parallelogram is a quadrilateral with all of the opposite sidesBeing parallel. Thus, all the sides are equal. Also, there are several other Properties in a parallelogram because the opposite sides are parallel.

Opposite sides areparallel

Rectangle: a rectangle is a parallelogram with right angles. Nothing more.

Right angles

Rhombus: a rhombus: take a parallelogram. Give it equal sides. Viola! A Rhombus.

Square: a square is a combination of a rectangle and a rhombus. It hasCongruent, parallel sides, the diagonals bisect each other and areCongruent, and the angles are right.

Big Square

The Opposite sides of a Parallelogram Theorem: this theorem says that theOpposite sides of a parallelogram, which are parallel, are of equal length. This is true because the diagonals of a parallelogram divide the parallelogramInto two congruent triangles, and since the corresponding parts of the Triangles are congruent, the sides are congruent.

Because this thing is a parallelogram, the opposite sides are congruent.

Given: ABCD is a parallelogram, EF,ADE CBF A

B C

D

E

F

Prove: ADEBCF

S R

1: Givens

2: AD=BC

3: ADEBCF

1: Given

2: Opposite sides of a ||gram

3: SAA Postulate

The Opposite angles of a ||gram Theorem

This theorem is similar to the last one: the opposite angles of a Parallelogram are equal in measure. This is easy to prove because ofThe diagonals of a parallelogram theorem: the diagonals of a ||gramDivide it into 2 congruent triangles. Then you can just take the ensuingcongruent triangles and compare the corresponding angles, which areCongruent.

congruent

Given: ABCD is a parallelogram

Prove: Theorem 9-14

A

B C

D

S R

1: ABCD is a ||gram

2: AD BC, AB CD

3: A C

4: ABD BCD

1:Given

2: Opp. Sides of ||gram

3: Opp. s of ||gram\

4: SAS Postulate