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