Download The Opposite sides of a Parallelogram Theorem

Chapter 9
Paul Hein
Period 2
12/12/2003
Parallelism
Key Terms
Skew Lines: Skew lines are 2 lines that are neither parallel nor intersect: thus, they must b
planes.
L1 and L2 are intersecting lines. L1 and L3 are parallel line
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 trans
Are on opposite sides of the transversal, are formed by 2 different coplanar lines, and are on
L1 and L2 are 2 coplanar lines. T is a transversal of them. A is an a
C, 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:
Cut by a transversal, then any two angles on the interior of the parallel lines and on the sam
Description.
(fig above) L1 and L2 are coplanar lines. T is a transversal of them. A is an interior
Angle 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 vertica
Angles is a corresponding angle to the other alternate interior angle.
L1 and L2 are coplanar, with transversal T. A is ve
B is an alternate interior angle to C. therefore, 
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.
Given: A is congruent to
B
Proof of AIP theorem
Prove: L1L2
L1
A
S
R
1. A B
1. Given
2. L1L2
2. AIP theorem
B
L2
The CAP theorem
The 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 are
Congruent, then the two lines are parallel.
T
A
If A is congruent to B,
Then L1 is parallel to L2.
L1
B
L2
Given: A is congruent to B
T
Prove: L1||L2
A
L1
S
1. AB
2. L1||L2
R
1.Given
2. CAP theorem
B
L2
Triangles
Key Terms
Right Triangle: A right triangle is a triangle with one right angle (90). Because of
This, we can conclude that the two other angles are acute, because all of the angles
In 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 properties
About 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.
Pythagorean Theorem: A²+B²=C²
Hypotenuse
C
A
B
Triangles
The angles of a triangle theorem*
This theorem states that all of the angles of a triangle add up to 180. There is no
Way to prove this theorem, but it is possible to prove that all of the angles of a
Triangle measure up to less than 181.
A
Given: A and B are complementary
Prove: C is right
S
1: A is comp. To B
2: ma +mb=90
3: ma+mb+mc=180
4: c+90=180
5: c=90
6: C is right
R
1: Given
2: Defn. of comp.
3: Angles of a triangle thm.
4: Substitution
5: Subtracti0n prop. Of =
6: Defn. of right angle
B
C
*Not real name
Acute angles of a Right triangle theorem
This 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 90
Degrees, that’s 90 off the 180 requirement. Thus, the other angles must add up to be
90 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=60
A
S
C
R
1: Givens
1: Given
2: a vert. To c
2: Defn. of Vert. 
3: ac
3: VAT
4: mc=30
4: Substitution
5: c comp. To d
5: acute s of a rt.
Thm
6: defn. of comp.
6: md=60
D
B
Quadrilaterals
Key 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.
P
Q
S
Base sides
medians
R
Trapezoid: a trapezoid is a quadrilateral with one and only one pair of sides
That are parallel. The parallel sides are called the base sides, and the
Nonparallel sides are called the medians.
Parallelogram: a parallelogram is a quadrilateral with all of the opposite sides
Being parallel. Thus, all the sides are equal. Also, there are several other
Properties in a parallelogram because the opposite sides are parallel.
Opposite sides are
parallel
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 has
Congruent, parallel sides, the diagonals bisect each other and are
Congruent, and the angles are right.
Big Square
The Opposite sides of a Parallelogram Theorem: this theorem says that the
Opposite sides of a parallelogram, which are parallel, are of equal length.
This is true because the diagonals of a parallelogram divide the parallelogram
Into 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.
E
Given: ABCD is a parallelogram, EF,
ADE CBF
A
D
Prove: ADEBCF
S
R
1: Givens
1: Given
2: AD=BC
2: Opposite sides of a ||gram
3: ADEBCF
3: SAA Postulate
B
C
F
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 of
The diagonals of a parallelogram theorem: the diagonals of a ||gram
Divide it into 2 congruent triangles. Then you can just take the ensuing
congruent triangles and compare the corresponding angles, which are
Congruent.
congruent
A
Given: ABCD is a parallelogram
D
Prove: Theorem 9-14
S
R
1: ABCD is a ||gram
1:Given
2: AD  BC, AB  CD
2: Opp. Sides of ||gram
3: A  C
3: Opp. s of ||gram\
4: ABD  BCD
4: SAS Postulate
B
C