Download Alternate Interior Angles

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.
•Parallel Lines: Any two coplanar lines that do not intersect.
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.
•Interior Angles on the Same Side of the Transversal: any two angles
that are on the same side of a transversal and are between the two lines
that are cut by the transversal.
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.
alternate interior angles and are congruent.
Conclusion: L1 and L2 are parallel.
If
Then
a and
b are
Given
EGB
CHF, prove that AB and CD are parallel.
Hypothesis:
EGB
Conclusion: AB
Statements
Reasons
1.
EGB
CHF
1.
given
2.
AGF
DHE
EGB,
CHF
2.
VAT
3.
substitution
3.
AGF
DHE
4.
AIP
4.
AB CD
CD.
CHF.
•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.
and b are corresponding angles.
Conclusion: a and
If
b are congruent.
Then
L1
L1
L2
L2
T
T
a
If BC=FG, CD=HG, and AC EG, prove
Statements
1.
BC=FG, CD=HG, AC EG
BCD
Reasons
1.
Given
2.
FGH
BCD
2.
PCA
3.
BCD
FHG
3.
SAS
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.
A
d is an exterior angle
of
ABC.
•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
Conclusion:
A+
ABC
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
Statements:
1.
z and
2.
m y+m
3.
m
4.
180=180
5.
m y+m
z=m
6.
m
w+m
w+m
z=m
y are supp
z=180
x+m
y=180
w+m
x
x
Reasons:
1.
Supp Post
2.
Def of supp
3.
Given a triangle, sum of
angles=180
x+m y 4.
5.
6.
Reflex ax of =
Trans ax of =
Sub ax of =
•Midline Theorem: a segment connecting the midpoints of two sides of a
triangle is parallel to the third side and has a length equal to half that of
the third side.
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, BC=GH, and D, E, I, and J are midpoints, prove
ADE
FIJ.
Reasons:
Statements:
1.
AB=FG, AC=FH, BC=GH, D, E, I,
J are midpoints
2.
DE=BC/2, IJ=GH/2
3.
BC/2=GH/2
4.
DE=IJ
5.
ADE
FIJ
1.
given
2.
midline theorem
3.
Div ax of =
4.
Substitution
5.
SSS
Quadrilaterals
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
•Convex Quadrilateral: Any two of the vertices of a convex quadrilateral
do not lie on opposite sides of a line that contains a segment of the
quadrilateral.
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:
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:
1.
•ABCD is a parallelogram
1.
Given
2.
DB=DB
2.
Reflex ax of =
3.
AD BC, AB CD
3.
Def of parallelogram
4.
ADB
ABD
CBD,
4.
PAI
5.
ASA
5.
ADB
CBD
6.
CPCTC
6.
A
C
CDB
•If two sides of a quadrilateral are parallel and congruent, it is a
parallelogram.
Restatement: Given•ABCD
where AB=CD and AB
CD.
Conclusion: •ABCD is a
parallelogram.
Hypothesis: AB=CD and
ABD
CDB.
Conclusion: •ABCD is a parallelogram.
Statements:
1.
AB=CD,
2.
3.
ABD
CDB
Reasons:
1.
Given
AB CD
2.
AIP
•ABCD is a parallelogram
3.
If two sides are parallel and congruent in a
quadrilateral, it is a parallelogram.