Download 2.3 Proving Lines Are Parallel

4.1 Quadrilaterals
Quadrilateral
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Isosceles
Trapezoid
4.1 Properties of a Parallelogram
• Definition: A parallelogram is a quadrilateral in
which both pairs of opposite sides are parallel.
AB || CD and BC || AD
D
A
B
C
4.1 Properties of a Parallelogram
• Properties of a parallelogram:
–
–
–
–
Opposite angles are congruent
Opposite sides are congruent
Diagonals bisect each other
Consecutive angles are supplementary
4.1 Properties of a Parallelogram
• In the following parallelogram:
AB = 7, BC = 4, mADC  63
A
D
–
–
–
–
What is CD?
What is AD?
What is mABC?
What is mDCB?
B
C
4.2 Proofs
• Proving a quadrilateral is a parallelogram:
– Show both pairs of opposite sides are parallel
(definition)
– Show one pair of opposite sides are congruent and
parallel
– Show both pairs of opposite sides are congruent
– Show the diagonals bisect each other
4.2 Kites
• Kite - a quadrilateral with two distinct pairs
of congruent adjacent sides.
• Theorem: In a kite, one pair of opposite
angles is congruent.
4.2 Midpoint Segments
• The segment that joins the midpoints of two sides of a
triangle is parallel to the third side and has a length equal to
½ the length of the third side.
MN  12 BC and MN || BC
A
M
C
N
B
4.3 Rectangle, Square, and Rhombus
• Rectangle - a parallelogram that has 4 right angles.
• The diagonals of a rectangle are congruent.
• A square is a rectangle that has all sides congruent
(regular quadrilateral).
4.3 Rectangle, Square, and Rhombus
• A rhombus is a parallelogram with all sides
congruent.
• The diagonals of a rhombus are perpendicular.
4.3 Rectangles: Pythagorean Theorem
• Pythagorean Theorem: In a right triangle, with the
hypotenuse of length c and legs of lengths a and b,
it follows that c2 = a2 + b2
a
c
b
Note: You can use this to get the length of the
diagonal of a rectangle.
4.4 The Trapezoid
• Definition: A trapezoid is a quadrilateral with
exactly 2 parallel sides.
Base
Leg
Leg
Base
Base angles
4.4 The Trapezoid
• Isosceles trapezoid:
– 2 legs are congruent
– Base angles are congruent
– Diagonals are congruent
4.4 The Trapezoid
• Median of a trapezoid:
connecting midpoints
of both legs
A
B
M
N
D
C
MN  12 ( AB  DC ) and AB || MN || DC
4.4 Miscellaneous Theorems
• If 3 or more parallel lines intercept congruent
segments on one transversal, then they intercept
congruent segments on any transversal.
5.1 Ratios, Rates, and Proportions
a
• Ratio sometimes written as a:b
b
Note: a and b should have the same units of measure.
a
• Rate b
like ratio except the units are different
(example: 50 miles per hour)
• Extended Ratio: Compares more than 2 quantities
example: sides of a triangle are in the ratio 2:3:4
5.1 Ratios, Rates, and Proportions
•
a c
Proportion - 
b d
two rates or ratios are equal
(read “a is to b as c is to d”)
a c
• Means-extremes property:
  ad  bc
b
d
product of the means = product of the extremes
where a,d are the extremes and b,c are the means
(a.k.a. “cross-multiplying”)
5.1 Ratios, Rates, and Proportions
•
a b
Geometric Mean -   b  ac
b c
b is the geometric mean of a & c
•
a c e
Extended Proportion s -  
b d f
…..used with similar triangles
5.1 Ratios, Rates, and Proportions
• Ratios – property 2:
a c
a b
d c
    
b d
c d
b a
(means and extremes may be switched)
• Ratios – property 3:
a c
ab cd
a b c d
 



b d
b
d
b
d
Note: cross-multiplying will always work, these
may lead to a solution faster sometimes
5.2 Similar Polygons
•
Definition: Two Polygons are similar 
two conditions are satisfied:
1. All corresponding pairs of angles are
congruent.
2. All corresponding pairs of sides are
proportional.
Note: “~” is read “is similar to”
5.2 Similar Polygons
•
Given ABC ~ DEF with the following
measures, find the lengths DF and EF:
E
10
5
A
B
D
6
4
C
F
5.3 Proving Triangles Similar
•
•
Postulate 15: If 3 angles of a triangle are
congruent to 3 angles of another triangle,
then the triangles are similar (AAA)
Corollary: If 2 angles of a triangle are
congruent to 2 angles of another triangle,
then the triangle, then the triangles are
similar. (AA)
5.3 Proving Triangles Similar
•
AA - If 2 angles of a triangle are congruent to 2
angles of another triangle, then the triangle, then
the triangles are similar.
•
SAS~ - If a an angle of one triangle is congruent
to an angle of a second triangle and the pairs of
sides including the two angles are proportional,
then the triangles are similar
5.3 Proving Triangles Similar
•
•
•
SSS~ - If the 3 sides of one triangle are
proportional to the three sides of another
triangle, then the triangles are similar
CSSTP – Corresponding Sides of Similar
Triangles are Proportional (analogous to
CPCTC in triangle congruence proofs)
CASTC – Corresponding angles of similar
triangles are congruent.
5.3 Proving Triangles Similar
• (example proof using CSSTP)
Statements
1. mA = m D
2. mB = m E
3. ABC ~ DEF
BC
4. AB

DE
EF
Reasons
1. Given
2. Given
3. AA
4. CSSTP
5.4 Pythagorean Theorem
• Pythagorean Theorem: In a right triangle, with the
hypotenuse of length c and legs of lengths a and b,
it follows that c2 = a2 + b2
c
a
b
• Converse of Pythagorean Theorem: If for a
triangle, c2 = a2 + b2 then the  opposite side c is
a right angle and the triangle is a right triangle.
5.4 Pythagorean Theorem
• Pythagorean Triples: 3 integers that satisfy the
Pythagorean theorem
–
–
–
–
3, 4, 5 (6, 8, 10; 9, 12, 15; etc.)
5, 12, 13
8, 15, 17
7, 24, 25
5.4 Classifying a Triangle by Angle
• If a, b, and c are lengths of sides of a triangle with
c being the longest,
c
– c2 > a2 + b2 
a
the triangle is obtuse
b
– c2 < a2 + b2 
the triangle is acute
– c2 = a2 + b2 
the triangle is right
5.5 Special Right Triangles
• 45-45-90 triangle:
– Leg opposite the 45 angle = a
– Leg opposite the 90 angle = 2a
45
a
2a
90
45
a
5.5 Special Right Triangles
• 30-60-90 triangle:
– Leg opposite 30 angle = a
– Leg opposite 60 angle = 3a
– Leg opposite 90 angle = 2a
60
2a
30
a
90
3a
5.6 Segments Divided Proportionally
• If a line is parallel to one side of a triangle and
intersects the other two sides, then it divides these
sides proportionally
AD
AE

DB
EC
AD
or

AE
AD
or

AB
A
DB
EC
AE
AC
D
B
E
C
5.6 Segments Divided Proportionally
• When 3 or more parallel lines are cut by a pair of
transversals, the transversals are divided
proportionally by the parallel lines
AB
DE

BC
EF
A
B
C
D
E
F
5.6 Segments Divided Proportionally
• Angle Bisector Theorem: If a ray bisects one angle
of a triangle, then it divides the opposite side into
segments whose lengths are proportional to the
length of the 2 sides which form that angle.
C
AC
CB

AD DB
A
D
B