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# Geometry Chapter 8

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by

Tweet## Christopher Cole

on 31 July 2012#### Transcript of Geometry Chapter 8

Chapter 8.1 Find Angle Measures in Polygons Polygon Interior Angles

Theorem Polygon Exterior Angles

Theorem Corollary to Theorem 8.1 The sum of the measures of the interior angles

of a convex n-gon is (n-2) x 180.

m<1 + m<2 + ... + m<n = (n-2) x 180 1 2 3 4 5 n = 5

What's the total measure of interior angles for this polygon? The sum of the measures of the interior angles of a quadrilateral is 360 degrees. Who can give me an example for the corollary? The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees.

m<1 + m<2 + ... + m<n = 360 degrees N = 5

What's measure of each exterior angle? Vocabulary In a polygon, two vertices that are endpoints of the same side are called consecutive vertices.

A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Polygon ABCDEF has three diagonals from vertex A, AC, AD and AE. diagonals Anyone??? Examples Shining Chapter 8.2 Use Properties of Parallelograms Vocabulary A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

The term "parallelogram DCBA" can be written as DCBA. In DCBA, DC II BA and AD II CB by definition. Theorem 8.3

(no-name) If a quadrilateral is a parallelogram, then its opposite sides are congruent.

If DCBA is a parellelogram, then DC = AB and

DA = CB. Theorem 8.4

(no-name) If a quadrilateral is a parallelogram, then its opposite angles are congruent.

If DCBA is a parellelogram, then <D = <B and

<C = <A. Theorem 8.5

(no-name) If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

If DCBA is a parallelogram, then

x + y = 180. 0 o o x y x y o o o o Theorem 8.6

(no-name) If a quadrilateral is a parellelogram, then its diagonals bisect each other.

DE = BE and CE = AE EXAMPLES Real Life Chapter 8.3 Show that a Quadrilateral

is a Paralellogram Theorem 8.7

(no-name) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If DC = BA and DA = CB, then DCBA is a parallelogram. Theorem 8.8

(no-name) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If <D = <B and <C = <A then DCBA is a parallelogram. Theorem 8.9

(no-name) If one pair of opposite sides of a quadrilateral are congurent and parallel, then the quadrilateral is a parallelogram.

If DC II to BA and DC = BA, then DCBA is a parallelogram. Theorem 8.10

(no-name) If the diagonals of a quadrialteral bisect each other, then the quadrilateral is a parallelogram.

If DB and CA bisect each other, then the quadrilateral is a parallelogram. Concept Summary Ways to Prove a Quadrilateral is a Parallelogram 1. 2. 3. 4. 5. Show both pairs of opposite sides are parallel. Show both pairs of opposite sides are congruent. Show both pairs of opposite angles are congruent. Show one pair of opposite sides are congruent and parallel. Show the diagonals bisect each other. examples Chapter 8.4 Properties of Rhombuses,

Rectangles, and Squares Vocabulary A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. Corollaries Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides.

ABCD is a rhombus if and only if

AB=BC=CD=AD. A B C D Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles.

ABCD is a rectangle if and only if

<A=<B=<C=<D. Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectanlge.

ABCD is a square if and only if

AB=BC=CD=AD and <A=<B=<C=<D are right angles. A B C D A B C D Theorems Theorem 8.11 A parallelogram is a rhombus if and only if its diagonals are perpendicular.

ABCD is a rhombus if and only if AC BD. Theorem 8.12 C B A D A parellelogram is a rhombus if and only if each diagonals bisects a pair of opposite angles.

ABCD is a rhombus if and only if AC bisects <BCD and <BAD and BD bisescts <ABC and <ADC. A B C D Theorem 8.13 A parallelogram is a rectangle if and only if its diagonals are congruent.

ABCD is a rectangle if and only if AC=BD. Now putting all of this together... What's the name of the yellow section here? examples Chapter 8.5 Use Properties of

Trapezoids and Kites Vocabulary A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. The nonparallel sides are the legs of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. The midsegement of a trapezoid connects the midpoints of its legs. A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Theorems Theorem 8.14 Theorem 8.15 Theorem 8.16 If a trapezoid is isosceles, then each pair of base angles is congruent. If trapezoid ABCD is isosceles,

then <A=<D and <B=<C. If the trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. If <A=<D (or if <B=<C), then

trapezoid ABCD is isosceles. A trapezoid is isosceles if and only if its diagonals are congruent. Trapezoid ABCD is isosceles if and only if AC=BD. Theorems Midsegment Theorem for Trapezoids The midsgement of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. If MN is the midsegment of the trapezoid ABCD, then MN AB, MN DC, and MN = 1/2 (AB+CD). A B C D A B C D A B C D A B C M N D Theorem 8.18 If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral ABCD is a kite, then AC BD. A B C D Theorem 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral ABCD is a kite and BC=BA, then <A=<C and <B=<D. A B C D Seriously...more of these??? Examples Let's work some of your favorite... Chapter 8.6 Identify Special

Quadrilaterals Also, one pair of congruent opposite <'s. THE END! What page is this in our book?

Full transcriptTheorem Polygon Exterior Angles

Theorem Corollary to Theorem 8.1 The sum of the measures of the interior angles

of a convex n-gon is (n-2) x 180.

m<1 + m<2 + ... + m<n = (n-2) x 180 1 2 3 4 5 n = 5

What's the total measure of interior angles for this polygon? The sum of the measures of the interior angles of a quadrilateral is 360 degrees. Who can give me an example for the corollary? The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 degrees.

m<1 + m<2 + ... + m<n = 360 degrees N = 5

What's measure of each exterior angle? Vocabulary In a polygon, two vertices that are endpoints of the same side are called consecutive vertices.

A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Polygon ABCDEF has three diagonals from vertex A, AC, AD and AE. diagonals Anyone??? Examples Shining Chapter 8.2 Use Properties of Parallelograms Vocabulary A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

The term "parallelogram DCBA" can be written as DCBA. In DCBA, DC II BA and AD II CB by definition. Theorem 8.3

(no-name) If a quadrilateral is a parallelogram, then its opposite sides are congruent.

If DCBA is a parellelogram, then DC = AB and

DA = CB. Theorem 8.4

(no-name) If a quadrilateral is a parallelogram, then its opposite angles are congruent.

If DCBA is a parellelogram, then <D = <B and

<C = <A. Theorem 8.5

(no-name) If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

If DCBA is a parallelogram, then

x + y = 180. 0 o o x y x y o o o o Theorem 8.6

(no-name) If a quadrilateral is a parellelogram, then its diagonals bisect each other.

DE = BE and CE = AE EXAMPLES Real Life Chapter 8.3 Show that a Quadrilateral

is a Paralellogram Theorem 8.7

(no-name) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If DC = BA and DA = CB, then DCBA is a parallelogram. Theorem 8.8

(no-name) If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If <D = <B and <C = <A then DCBA is a parallelogram. Theorem 8.9

(no-name) If one pair of opposite sides of a quadrilateral are congurent and parallel, then the quadrilateral is a parallelogram.

If DC II to BA and DC = BA, then DCBA is a parallelogram. Theorem 8.10

(no-name) If the diagonals of a quadrialteral bisect each other, then the quadrilateral is a parallelogram.

If DB and CA bisect each other, then the quadrilateral is a parallelogram. Concept Summary Ways to Prove a Quadrilateral is a Parallelogram 1. 2. 3. 4. 5. Show both pairs of opposite sides are parallel. Show both pairs of opposite sides are congruent. Show both pairs of opposite angles are congruent. Show one pair of opposite sides are congruent and parallel. Show the diagonals bisect each other. examples Chapter 8.4 Properties of Rhombuses,

Rectangles, and Squares Vocabulary A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. Corollaries Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides.

ABCD is a rhombus if and only if

AB=BC=CD=AD. A B C D Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles.

ABCD is a rectangle if and only if

<A=<B=<C=<D. Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectanlge.

ABCD is a square if and only if

AB=BC=CD=AD and <A=<B=<C=<D are right angles. A B C D A B C D Theorems Theorem 8.11 A parallelogram is a rhombus if and only if its diagonals are perpendicular.

ABCD is a rhombus if and only if AC BD. Theorem 8.12 C B A D A parellelogram is a rhombus if and only if each diagonals bisects a pair of opposite angles.

ABCD is a rhombus if and only if AC bisects <BCD and <BAD and BD bisescts <ABC and <ADC. A B C D Theorem 8.13 A parallelogram is a rectangle if and only if its diagonals are congruent.

ABCD is a rectangle if and only if AC=BD. Now putting all of this together... What's the name of the yellow section here? examples Chapter 8.5 Use Properties of

Trapezoids and Kites Vocabulary A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. The nonparallel sides are the legs of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. The midsegement of a trapezoid connects the midpoints of its legs. A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Theorems Theorem 8.14 Theorem 8.15 Theorem 8.16 If a trapezoid is isosceles, then each pair of base angles is congruent. If trapezoid ABCD is isosceles,

then <A=<D and <B=<C. If the trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. If <A=<D (or if <B=<C), then

trapezoid ABCD is isosceles. A trapezoid is isosceles if and only if its diagonals are congruent. Trapezoid ABCD is isosceles if and only if AC=BD. Theorems Midsegment Theorem for Trapezoids The midsgement of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. If MN is the midsegment of the trapezoid ABCD, then MN AB, MN DC, and MN = 1/2 (AB+CD). A B C D A B C D A B C D A B C M N D Theorem 8.18 If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral ABCD is a kite, then AC BD. A B C D Theorem 8.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral ABCD is a kite and BC=BA, then <A=<C and <B=<D. A B C D Seriously...more of these??? Examples Let's work some of your favorite... Chapter 8.6 Identify Special

Quadrilaterals Also, one pair of congruent opposite <'s. THE END! What page is this in our book?