4 1 quadrilaterals
Download
1 / 32

4.1 Quadrilaterals - PowerPoint PPT Presentation


  • 350 Views
  • Uploaded on

4.1 Quadrilaterals. Quadrilateral. Parallelogram. Trapezoid. Isosceles Trapezoid. Rhombus. Rectangle. Square. 4.1 Properties of a Parallelogram. Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A. B. D. C.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about '4.1 Quadrilaterals' - ocean


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
4 1 quadrilaterals
4.1 Quadrilaterals

Quadrilateral

Parallelogram

Trapezoid

IsoscelesTrapezoid

Rhombus

Rectangle

Square


4 1 properties of a parallelogram
4.1 Properties of a Parallelogram

  • Definition: A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

A

B

D

C


4 1 properties of a parallelogram1
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 parallelogram2
4.1 Properties of a Parallelogram

  • In the following parallelogram:AB = 7, BC = 4,

    • What is CD?

    • What is AD?

    • What is mABC?

    • What is mDCB?

A

B

D

C


4 2 proofs
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
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
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

M

N

B

C


4 3 rectangle square and rhombus
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 rhombus1
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
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 + b2Note: You can use this to get the length of the diagonal of a rectangle.

a

c

b


4 4 the trapezoid
4.4 The Trapezoid

  • Definition: A trapezoid is a quadrilateral with exactly 2 parallel sides.

Base

Leg

Leg

Base

Base angles


4 4 the trapezoid1
4.4 The Trapezoid

  • Isosceles trapezoid:

    • 2 legs are congruent

    • Base angles are congruent

    • Diagonals are congruent


4 4 the trapezoid2
4.4 The Trapezoid

A

B

  • Median of a trapezoid:connecting midpointsof both legs

M

N

D

C


4 4 miscellaneous theorems
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
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


5 1 ratios rates and proportions1
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”)


5 1 ratios rates and proportions2
5.1 Ratios, Rates, and Proportions

  • b is the geometric mean of a & c

  • …..used with similar triangles


5 1 ratios rates and proportions3
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


5 2 similar polygons
5.2 Similar Polygons

  • Definition: Two Polygons are similar  two conditions are satisfied:

    • All corresponding pairs of angles are congruent.

    • All corresponding pairs of sides are proportional.Note: “~” is read “is similar to”


5 2 similar polygons1
5.2 Similar Polygons

  • Given ABC ~ DEF with the following measures, find the lengths DF and EF:

E

10

5

B

D

A

6

4

C

F


5 3 proving triangles similar
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 similar1
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 similar2
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 similar3
5.3 Proving Triangles Similar

  • (example proof using CSSTP)


5 4 pythagorean theorem
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.

c

a

b


5 4 pythagorean theorem1
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
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

c

a

b


5 5 special right triangles
5.5 Special Right Triangles

  • 45-45-90 triangle:

    • Leg opposite the 45 angle = a

    • Leg opposite the 90 angle =

45

a

90

45

a


5 5 special right triangles1
5.5 Special Right Triangles

  • 30-60-90 triangle:

    • Leg opposite 30 angle = a

    • Leg opposite 60 angle =

    • Leg opposite 90 angle = 2a

60

2a

a

30

90


5 6 segments divided proportionally
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

C


5 6 segments divided proportionally1
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

B

E

C

F


5 6 segments divided proportionally2
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

A

B

D


ad