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# 4.1 Quadrilaterals - PowerPoint PPT Presentation

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.

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Presentation Transcript

Parallelogram

Trapezoid

IsoscelesTrapezoid

Rhombus

Rectangle

Square

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

A

B

D

C

• Properties of a parallelogram:

• Opposite angles are congruent

• Opposite sides are congruent

• Diagonals bisect each other

• Consecutive angles are supplementary

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

• What is CD?

• What is mABC?

• What is mDCB?

A

B

D

C

• 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

• Kite - a quadrilateral with two distinct pairs of congruent adjacent sides.

• Theorem: In a kite, one pair of opposite angles is congruent.

• 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

• 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).

• A rhombus is a parallelogram with all sides congruent.

• The diagonals of a rhombus are perpendicular.

• 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

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

Base

Leg

Leg

Base

Base angles

• Isosceles trapezoid:

• 2 legs are congruent

• Base angles are congruent

• Diagonals are congruent

A

B

• Median of a trapezoid:connecting midpointsof both legs

M

N

D

C

• If 3 or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any transversal.

• 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

• 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”)

• b is the geometric mean of a & c

• …..used with similar triangles

• 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

• 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”

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

E

10

5

B

D

A

6

4

C

F

• 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)

• 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

• 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.

• (example proof using CSSTP)

• 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

• 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

• 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

• 45-45-90 triangle:

• Leg opposite the 45 angle = a

• Leg opposite the 90 angle =

45

a

90

45

a

• 30-60-90 triangle:

• Leg opposite 30 angle = a

• Leg opposite 60 angle =

• Leg opposite 90 angle = 2a

60

2a

a

30

90

• 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

• 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

• 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