4.1 quadrilaterals quadrilateral parallelogramtrapezoid rectangle rhombus square isosceles trapezoid
TRANSCRIPT
4.1 Quadrilaterals
Quadrilateral
Parallelogram Trapezoid
Rectangle Rhombus
Square
IsoscelesTrapezoid
4.1 Properties of a Parallelogram
• Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A B
CD
ADBCandCDAB ||||
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,
– What is CD?
– What is AD?
– What is mABC?
– What is mDCB?
A B
CD
63ADCm
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.
A
BC
M N
BCMNandBCMN ||21
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
Note: You can use this to get the length of the diagonal of a rectangle.
a
b
c
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 midpointsof both legs
M N
D C
BA
DCMNABandDCABMN ||||)(21
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
• Ratio - sometimes written as a:b
Note: a and b should have the same units of measure.
• Rate - like ratio except the units are different (example: 50 miles per hour)
• Extended Ratio: Compares more than 2 quantitiesexample: sides of a triangle are in the ratio 2:3:4
b
a
b
a
5.1 Ratios, Rates, and Proportions
•
two rates or ratios are equal (read “a is to b as c is to d”)
• Means-extremes property:
product of the means = product of the extremeswhere a,d are the extremes and b,c are the means(a.k.a. “cross-multiplying”)
d
c
b
a - Proportion
cbdad
c
b
a
5.1 Ratios, Rates, and Proportions
•
b is the geometric mean of a & c
•
…..used with similar triangles
acbc
b
b
a -Mean Geometric
f
e
d
c
b
a - sProportion Extended
5.1 Ratios, Rates, and Proportions
• Ratios – property 2: (means and extremes may be switched)
• Ratios – property 3:
Note: cross-multiplying will always work, these may lead to a solution faster sometimes
a
c
b
d
d
b
c
a
d
c
b
a
d
dc
b
ba
d
dc
b
ba
d
c
b
a
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:
A
C
6
D
F
E
5
4
B10
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 Reasons
1. mA = m D 1. Given
2. mB = m E 2. Given
3. ABC ~ DEF 3. AA
4. 4. CSSTPEF
BC
DE
AB
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
• 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.
a
b
c
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,– c2 > a2 + b2
the triangle is obtuse– c2 < a2 + b2
the triangle is acute– c2 = a2 + b2
the triangle is right
a
b
c
5.5 Special Right Triangles
• 45-45-90 triangle:– Leg opposite the 45 angle = a
– Leg opposite the 90 angle =
a
a
a2
a2
45
90 45
5.5 Special Right Triangles
• 30-60-90 triangle:– Leg opposite 30 angle = a
– Leg opposite 60 angle =
– Leg opposite 90 angle = 2a
a2a
a390
a3
30
60
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
A
D E
B CAC
AE
AB
ADor
EC
DB
AE
ADor
EC
AE
DB
AD
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
A D
EB
C F
EF
DE
BC
AB
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.
AD B
C
DB
CB
AD
AC