Geometry Unit 6 Properties of Quadrilaterals Classifying ...mathwithjp.weebly.com/.../0/...quadrilateral_notes.pdf · quadrilateral are congruent, then the quadrilateral is a parallelogram.

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Geometry Unit 6 Properties of Quadrilaterals

Classifying Polygons Review

Polygon – a closed plane figure with at least 3 sides that are segments

-the sides do not intersect except at the vertices

N-gon - a polygon with n-sides.

Diagonal – a segment joining two non-adjacent vertices of a polygon.

Regular Polygon – A polygon where all sides and all angles are congruent.

Convex Polygon – has no diagonal with points outside the polygon

Concave Polygon – has at least one diagonal with points outside the polygon

Type ________________ _________________ ________________________

Classify ________________ _________________ ________________________

Classifying a polygons by sides:

3 sides - 6 sides - 9 sides -

4 sides - 7 sides - 10 sides -

5 sides - 8 sides - n sides -

Naming a polygon - A polygon is named by starting at any vertex and going clockwise or

counterclockwise to the next consecutive vertex.

This polygon can be named two

different ways starting at point M:

1) Polygon M_________

2) Polygon M_________

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6-0 Interior and Exterior Angles of Polygons

How many triangles can be drawn in this square by drawing in diagonals (diagonals cannot intersect)

*The sum of the interior angles of a triangle is 180

The sum of the interior angles of a square is180 + 180 = 360

Investigate the number of triangles that can be drawn in the following shapes with diagonals:

______ Triangles ______ Triangles ______ Triangles

Sum = 180*__=_____ Sum = 180*__=_____ Sum = 180*__=_____

*There are always 2 less triangles than sides in every convex polygon.

Polygon-Angle Sum Theorem

The sum of the measures of the interior angles of a convex n-gon is

Triangle = (3 – 2) * 180 Pentagon = _____________

= 1 * 180 = _________

= 180 = _______

Finding the missing angle value

This shape at right is a pentagon.

The interior angle sum is (5 – 2)*180 = 540

The sum of the angles currently is (437 + x)

Substituting we have: (437 + x) = 540

Solving for x gives the answer x = 103

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What is the sum of the measures of the exterior angles of an equilateral triangle and of a rectangle?

Polygon Exterior Angle Sum Theorem

The sum of the measures of the exterior angles of any polygon is 360

***Special rules for REGULAR polygons***

All of the interior angles are congruent ***After you find the measure of the interior angles, divide that number by the # of

vertices to find the measure of each vertex angle.

All of the exterior angles are congruent

Divide 360 by the number of vertices to find the measure of each exterior angle.

, *n is the number of sides of the polygon.

To find each interior angle of a regular polygon then divide by the number of sides.

*n is the number of sides of the polygon

What is the exterior angle of a regular polygons?

Hexagon =

Pentagon = _____________

= = _________

What is the measeure of the interior angles of the regular polygons?

Octagon =

Dodecagon = _____________

=

= _________

= 135 = _______

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6-1 Classifying Quadrilaterals

Parallelogram Rectangle

Square Kite

Trapezoiod Isosceles Trapezoid

Rhombus

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6-2 Properties of Parallelograms

All properties on this page are true when a quadrilateral is a parallelogram

Parallelogram: Is a quadrilateral with both pairs of opposite sides are parallel.

Parallelogram Opposite Side Congruence Theorem:

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Parallelogram Consecutive Interior Angle Theorem:

If a quadrilateral is a parallelogram, its consecutive angles are supplementary.

Parallelogram Opposite Angle Congruence Theorem:

If a quadrilateral is a parallelogram, its opposite angles are congruent.

Parallelogram Diagonal Bisector Theorem:

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Congruent Segments Theorem For Parallel Lines.

If three (or more) parallel lines make congruent

segments on one transversal, then they make

congruent segments on all transversals.

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6-3 Proving a quadrilateral is a parallelogram

Converse to Parallelogram Opposite Side Congruence Theorem: If both pairs of opposite sides of a

quadrilateral are congruent, then the quadrilateral is a parallelogram.

Converse to Parallelogram Consecutive Interior Angles Theorem: If an angle of a quadrilateral is

supplementary to both consecutive angles, then the quadrilateral is a parallelogram.

Converse to Parallelogram Opposite Angle Congruence Theorem: If both pairs of opposite angles of a

quadrilateral are congruent, then the quadrilateral is a parallelogram.

Converse to Parallelogram Diagonal Bisector Theorem: If the diagonals of a quadrilateral bisect each

other, the quadrilateral is a parallelogram.

Parallelogram Congruent and Parallel Congruence Theorem: If one pair of opposite sides of a

quadrilateral is both congruent and parallel, the quadrilateral is a parallelogram.

Proving a Quadrilateral is a Parallelogram

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Write a developmental proof for each of the following.

1. Given: AB DC and AD BC

Prove: ABCD is a Parallelogram

2. Given: and AB DC Prove: ABCD is a Parallelogram

C

A B

D

C

A B

D

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3. Given: 180A B and 180B C

Prove: ABCD is a Parallelogram

4. Given: AE EC and BE ED

Prove: ABCD is a Parallelogram

C

A B

D

E

C

A B

D

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5. Given: DAB BCD and ABC CDA

Prove: ABCD is a Parallelogram

C

A B

D

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6-4 Special Classes of Parallelograms

All Parallelogram rules apply as well as each shapes unique rules.

Rhombus: Is a parallelogram with all four sides congruent.

Rhombus Perpendicular Diagonal Theorem: A parallelogram is a rhombus if and only if its diagonals are

Perpendicular.

Rhombus Diagonals Bisect Vertex Angle Theorem: A parallelogram is a rhombus if and only if each

diagonal bisects a pair of opposite angles.

Rectangle: Is a parallelogram with four right angles.

Rectangle Diagonal Congruence Theorem: A parallelogram is a rectangle if and only if its diagonals of

are congruent.

Square: Is a rectangle that is also a rhombus.

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Z

L

J K

M

Z

L

J K

M

Developmental Proofs

Given: JKLM is a rhombus

Prove:

Given: JKLM is a parallelogram

at Z Prove: JKLM is a rhombus

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6-5 Trapezoids and Kites

1. Trapezoid: Is a quadrilateral with exactly one pair of parallel sides.

2. Isosceles Trapezoid: If the legs of the trapezoid are congruent.

3. Trapezoid Base Angles Congruence Theorem: If a trapezoid is isosceles, the each pair of base angles

is congruent.

4. Trapezoid Base Angles Congruent Theorem Converse: If a trapezoid has a pair of congruent base

angles, then it is an isosceles trapezoid.

5. Trapezoid Congruent Diagonals Theorem: A trapezoid is isosceles if and only if its diagonals are

congruent.

6. Trapezoid Midsegment Theorem: 1) The midsegment of a trapezoid is half the sum of the bases

2) The midsegment of a trapezoid is parallel to the bases of the

trapezoid.

7. Kite: Is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not

congruent.

8. Kite Perpendicular Diagonals Theorem: If a quadrilateral is a kite, then its diagonals are

perpendicular.

9. Kite One Opposite Congruent Angle Theorem: If a quadrilateral is a kite, then exactly one pair of

opposite angles are congruent.

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S A

H

W

T

Developmental Proofs

Given: WHAT is a kite, with Prove: