Download alternate interior angles

Parallelism  Triangles 
Quadrilaterals
Key Terms
skew lines: non-coplanar lines that do not intersect
(lines AB and EF are skew lines)
parallel lines: non-intersecting coplanar lines (lines AD
and EF are parallel lines)
C
B
A
D
E
F
transversal: line that intersects two coplanar lines (line AB is
a transversal)
alternate interior angles: angles that lie inside two lines and
on opposite sides of the transversal (r and s are alternate
interior angles)
corresponding angles: if r and s are alternate interior
angles and q is a vertical angle to r, then q and s are
corresponding angles.
B
r
A
s
q
Theorems
AIP Theorem
If you are given two lines intersected by a transversal, and
a pair of alternate interior angles are congruent, then the
lines are parallel.
Restatement: Given line AB and line CD cut by
transversal EF. If x  y, then line AB is parallel to line
CD.
A
F
B
x
C
y
D
E
Given: line segment GH and line segment JK bisect each
other at F.
Prove: line segment GK and line segment JH are parallel.
J
Statements
Reasons
1. GH and JK bisect at F
1. Given
2. GF = FH and JF = FK
2. Def. of bisector
3. JFH  KFG
3. VAT
4. ∆JFH  ∆KFG
4. SAS
5. HJF  GKF
5. CPCTC
6. JH is parallel to GK
6. AIP
H
F
G
K
PCA Corollary
Corresponding angles are congruent if you are given two
parallel lines cut by a transversal.
Restatement: v  w if line AB and line CD are parallel
and are cut by transversal EF.
F
A
C
v
w
E
B
D
Given: the figure with CDE  A and line LF  line AB.
Prove: line LF  line DE.
Statements
Reasons
C
1. CDE  A, LF  AB
1. Given
2. DE is parallel to AB
2. CAP
3. GFA  LGD
3. PCA
4. GFA is a R.A.
4. Def. of perp.
5. m GFA = 90˚
5. Def. of R.A.
6. m LGD = 90˚
6. Def. of congruence
7. LGD = R.A.
7. Def. of R.A.
8. LF  DE
8. Def. of perp.
L
G
D
E
H
A
B
Key Terms
concurrent lines: two or more lines that all share a common point
(lines AB, CD, and EF are concurrent.)
point of concurrency: the common point shared by concurrent
lines. (point G is the point of concurrency.)
E
A
D
G
C
B
F
Theorems
Theorem 9-13
The measures of all the angles in a triangle add up to 180.
Restatement: Given ABC. mA + m B +m C = 180.
B
A
C
Given: ABC, BA  AC and mB = 65.
Prove: mC = 155.
Statements
Reasons
1. BA  AC, m B = 65
1. Given
2. A is a R.A.
2. Def. of perp.
3. mA = 90
3. Def. of R.A.
4. mA + mB +mC = 180
4. ms in  = 180
5. 90 + 65 + mC = 180
5. Sub.
6. mC = 155
6. SPE
B
65˚
A
C
Theorem 9-28
If one side of a right triangle is half the length of the
hypotenuse, then the measure of the opposite angle is
30.
Restatement: Given right triangle ABC. If AB =
1/2BC, then mBCA is 30.
B
A
C
Given: DEF is a right triangle. D = 90 and DE = 1/2EF.
Prove: mE = 60
Statements
Reasons
E
1. DEF is a R.T.
DE = 1/2EF
2. mF = 30
3. mD = 90
4. mD + mE +
mF = 180
5. 90 + mE + 30 =
180
6. mE = 60
1. Given
2.  opp. side 1/2
as long as hyp. =
30
3. Def. of R.T.
4. s add up to
180
5. Sub.
6. SPE
F
D
Key Terms
diagonal: a line segment connecting two nonconsecutive angles
in a quadrilateral. (segment AC is a diagonal)
parallelogram: quadrilateral that has opposite parallel lines.
(•ABCD is a parallelogram)
trapezoid: quadrilateral that has one pair of opposite parallel
lines and one pair of nonparallel lines. (•JKLM is a trapezoid.)
B
A
C
D
K
L
J
M
rhombus: parallelogram that has 4 congruent sides. (•QRST
is a rhombus)
rectangle: parallelogram with 4 right angles. (•ABCD is a
rectangle)
square: rectangle that has 4 congruent sides. (•FGHJ is a
square)
R
Q
G
S
B
C
F
T
A
H
D
J
Theorems
Theorem 9-21
If the diagonals in a quadrilateral bisect each other, that
quadrilateral is a parallelogram.
Restatement: Given •ABCD. If AC and BD bisect each other
at E, then •ABCD is a parallelogram.
B
C
E
A
D
Given:•WXYZ WT = TY and XT = TZ
Prove:•WXYZ is a parallelogram.
Statements
Reasons
X
1. WT = TY, XT = TZ
1. Given
2. WY and XZ bisect
each other.
2. Def. of
bisectors
3.•WXYZ is a
parallelogram.
3. If diagonals
bisect each
other, =
parallelogram
Y
T
W
Z
Theorem 9-24
A rhombus’ diagonals are perpendicular to each other.
Restatement: Given •ABCD is a rhombus, then AC  BD.
B
A
C
D
Given:•FGHK is a rhombus.
Prove: GJF  GJH
Statements
Reasons
1.•FGHK is a rhombus
1. Given
2. GK  FH
2. Diagonals are  in rhombus
3. GJK and GJH are
R.A.
3. Def. of perp.
4. R.A. 
4. GJK  GJH
G
J
F
K
H
 

 
 
  