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Cross Product Property of proportions

Reciprocal property of proportions

Switching Property of proportions

Add property of proportions

The product of the extremes equals the product of the means.…

If 2 ratios are equal then their reciprocals are also equal.…

If a/b=c/d, then a/c=b/d

If a/b=c/d, then a+b/b=c+d/d

Cross Product Property of proportions

The product of the extremes equals the product of the means.…

Reciprocal property of proportions

If 2 ratios are equal then their reciprocals are also equal.…

geometric mean

simplest form radicals

common Pythagorean triples (5)

if the altitude is drawn to the hypote…

if a, b, and x are positive numbers and a/x = x/b , then x is…

1. No perfect square factor other than 1 is under the radical…

1. 3 - 4- 5... 2. 5 - 12 - 13 ... 3. 8 - 15 - 17... 4. 7 - 24 - 25... 5.…

then the 2 triangles formed are similar to the original trian…

geometric mean

if a, b, and x are positive numbers and a/x = x/b , then x is…

simplest form radicals

1. No perfect square factor other than 1 is under the radical…

Perimeters of Similar Polygons

Areas of Similar Polygons

Angle-Angle (AA) Similarity Theorem

Side-Side-Side (SSS) Similarity Theorem

If two polygons are similar, then the ratio of their perimete…

If two polygons are similar, then the ratio of their areas is…

If two angles of one triangle are... congruent to two angles of…

If the corresponding side lengths of... two triangles are propor…

Perimeters of Similar Polygons

If two polygons are similar, then the ratio of their perimete…

Areas of Similar Polygons

If two polygons are similar, then the ratio of their areas is…

*** theorem 8-1 ***

*** corollary 1 to theorem 8-1 ***

*** corollary 2 to theorem 8-1 ***

*** theorem 8-2 : pythagorean theorem ***

if the altitude is drawn to the hypotenuse of a right triangl…

when the altitude is drawn to the hypotenuse of a right trian…

when the altitude is drawn to the hypotenuse of a right trian…

in a right triangle , the square of the hypotenuse is equal t…

*** theorem 8-1 ***

if the altitude is drawn to the hypotenuse of a right triangl…

*** corollary 1 to theorem 8-1 ***

when the altitude is drawn to the hypotenuse of a right trian…

Interior Angle Theorem

Exterior Angle Theorem

Theorem 8.3

Theorem 8.4

If a convex polygon has n sides and S is the sum of the measu…

If a polygon is convex, then the sum of the measures of the e…

Opposite sides of parallelogram are congruent.

Opposite angles of a parallelogram are congruent.

Interior Angle Theorem

If a convex polygon has n sides and S is the sum of the measu…

Exterior Angle Theorem

If a polygon is convex, then the sum of the measures of the e…

Diagonal

Polygon Interior Angles Theorem

Interior Angles of a quadrilateral

Polygon Exterior Angles Theorem

segment that joins two nonconsecutive vertices of a polygon

The sum of the measures of the interior angles of a convex n-…

The sum of the measures of the interior angles of a quadrilat…

the sum of the measures of the exterior angles of a convex po…

Diagonal

segment that joins two nonconsecutive vertices of a polygon

Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n-…

means-extremes product theorem

similar polygons definition

perimeter theorem

3 methods of proving triangles congruent

product of means is equal to the product of extremes

1. corresponding angles are congruent... 2. corresponding sides…

the ratio of the perimeters of any 2 similar polygons equals…

1. AA similarity... 2. SSS similarity... 3. SAS similarity

means-extremes product theorem

product of means is equal to the product of extremes

similar polygons definition

1. corresponding angles are congruent... 2. corresponding sides…

Area postulate

Uniqueness Property

Congruence Property

Additive Property

Uniqueness, congruence, and additive properties and rectangle…

Given a unit region, every polygonal region has a unique area

Congruent figures have the same area

The area of the Union of two non-overlapping regions is the s…

Area postulate

Uniqueness, congruence, and additive properties and rectangle…

Uniqueness Property

Given a unit region, every polygonal region has a unique area

Postulate 1-1 Through any two points,…

Postulate 1-2 If two distinct lines in…

Postulate 1-3 If two distinct _______i…

Postulate 1-4 Through any ________poin…

line

point

planes

three noncollinear

Postulate 1-1 Through any two points,…

line

Postulate 1-2 If two distinct lines in…

point

Thm. 8-1-1

Geometric Means Corollary Pt.1

Geometric Means Corollary Pt.2

The Law of Sines

The altitude to the hypotenuse of a right triangle forms 2 tr…

The length of the altitude to the hypotenuse of a right trian…

The length of a leg of a right triangle is the geometric mean…

For any triangle ABC with side lengths a, b, and c:

Thm. 8-1-1

The altitude to the hypotenuse of a right triangle forms 2 tr…

Geometric Means Corollary Pt.1

The length of the altitude to the hypotenuse of a right trian…

Theorum 8.1: Polygon interior angles t…

Corollary to the Theorem 8.1: interior…

Theorem 8.2: Polygon Exterior Angles T…

Theorem 8.3

The sum of the measures of interior angles of a n-gon is (n-2…

The sum of the measures of interior angles of a quadrilateral…

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

If a quadrilateral is a parallelogram, then it's opposite sid…

Theorum 8.1: Polygon interior angles t…

The sum of the measures of interior angles of a n-gon is (n-2…

Corollary to the Theorem 8.1: interior…

The sum of the measures of interior angles of a quadrilateral…

The product of the extremes equals the…

If two ratios are equal, then their re…

If A/B = C/D , then A/C = B/D .

Two polygons are________if their corre…

Cross Product Property

Reciprocal Property

Interchange Means Property

Similar

The product of the extremes equals the…

Cross Product Property

If two ratios are equal, then their re…

Reciprocal Property

Perimeters of Similar Polygons

Areas of Similar Polygons

Angle-Angle Similarity Theorem

Side-Side-Side Similarity Theorem (SSS)

If 2 polygons are similar, then the ratio of their perimeters…

If two polygons are similar, then the ratios of their areas i…

If 2 angles of one triangle are congruent to 2 angles of anot…

If the corresponding sides lengths of two triangles are propo…

Perimeters of Similar Polygons

If 2 polygons are similar, then the ratio of their perimeters…

Areas of Similar Polygons

If two polygons are similar, then the ratios of their areas i…

Theorem 8-1: The Pythagorean Theorem

Theorem 8-2: Converse of the Pythagore…

Theorem 8-3

Theorem 8-4

If a triangle is a right triangle, then a² + b² = c².

If a² + b² = c², then the triangle is a right triangle.

If c² > a² + b², then the triangle is obtuse.

If c² < a² + b², then the triangle is acute.

Theorem 8-1: The Pythagorean Theorem

If a triangle is a right triangle, then a² + b² = c².

Theorem 8-2: Converse of the Pythagore…

If a² + b² = c², then the triangle is a right triangle.

Pythagorean Theorem

Pythagorean Theorem (alternate stateme…

Pythagorean Converse Theorem

Isosceles Right Triangle Theorem (a.k.…

In any right triangle with legs of lengths a and b and hypote…

In any right triangle, the sum of the areas of the squares on…

Suppose a triangle has sides of lengths a, b, and c. If a² +…

In any isosceles right triangle, if a leg has length x, then…

Pythagorean Theorem

In any right triangle with legs of lengths a and b and hypote…

Pythagorean Theorem (alternate stateme…

In any right triangle, the sum of the areas of the squares on…

Pythagorean Theorem

Pythagorean Theorem (alternate stateme…

Pythagorean Converse Theorem

Isosceles Right Triangle Theorem (a.k.…

In any right triangle with legs of lengths a and b and hypote…

In any right triangle, the sum of the areas of the squares on…

Suppose a triangle has sides of lengths a, b, and c. If a² +…

In any isosceles right triangle, if a leg has length x, then…

Pythagorean Theorem

In any right triangle with legs of lengths a and b and hypote…

Pythagorean Theorem (alternate stateme…

In any right triangle, the sum of the areas of the squares on…

Definition of Perimeter

Area Postulate

Right Triangle Angle Formula

Triangle Area Formula

The perimeter of a polygon is the sum of the lengths of its s…

a. Uniqueness Property... Given a unit region, every polygonal…

The area of a right triangle is half the product of the of th…

The area of a triangle is half the product of the length of a…

Definition of Perimeter

The perimeter of a polygon is the sum of the lengths of its s…

Area Postulate

a. Uniqueness Property... Given a unit region, every polygonal…

Perimeters of Similar Polygons

Areas of Similar Polygons

Angle-Angle (AA) Similarity Theorem

Side-Side-Side (SSS) Similarity Theorem

If two polygons are similar, then the ratio of their perimete…

If two polygons are similar, then the ratio of their areas is…

If two angles of one triangle are... congruent to two angles of…

If the corresponding side lengths of... two triangles are propor…

Perimeters of Similar Polygons

If two polygons are similar, then the ratio of their perimete…

Areas of Similar Polygons

If two polygons are similar, then the ratio of their areas is…

radius (regular polygon)

polygon

remote interior angles

linear scale factor

a line segment that connects the center of a regular polygon…

a two-dimensional closed figure of three or more line segment…

when a triangle has an exterior angles, these angles are the…

the ratio of similarity. the ratio of any pair of correspondi…

radius (regular polygon)

a line segment that connects the center of a regular polygon…

polygon

a two-dimensional closed figure of three or more line segment…

Parallel Postulate

Perpendicular Postulate

Corresponding Angles Theorem

Alternate Interior Angles Theorem

If there is a line and a point not on the line, then there is…

If there is a line and a point not on the line, then there is…

If two parallel lines are cut by a transversal, then the pair…

If two parallel lines are cut by a transversal, then the pair…

Parallel Postulate

If there is a line and a point not on the line, then there is…

Perpendicular Postulate

If there is a line and a point not on the line, then there is…

Postulate 3-1 Same-side Interior Angle…

Theorem 3-1 Alternate Interior Angles…

Theorem 3-2 Corresponding Angles Theor…

Theorem 3-3 Alternate Exterior Angles…

supplementary... m∠1 + m∠2 = 180

congruent

corresponding

exterior

Postulate 3-1 Same-side Interior Angle…

supplementary... m∠1 + m∠2 = 180

Theorem 3-1 Alternate Interior Angles…

congruent

Conclusion

Conditional Statement

Conjecture

Contrapositive

In a conditional statement (if-then statement), the statement…

A statement written in an if then form... A statement of the for…

An educated guess based on known information... pg 78

The statement formed by negating both the hypothesis(educated…

Conclusion

In a conditional statement (if-then statement), the statement…

Conditional Statement

A statement written in an if then form... A statement of the for…

Theorem 8-1-1

Geometric Means (Corollary 8-1-2) (Ign…

Geometric Means (Corollary 8-1-3) (Ign…

Trigonometric Ratios (Sine)

The altitude to the hypotenuse of a right triangle forms two…

The length of the altitude to the hypotenuse of a right trian…

The length of a leg of a right triangle is the geometric mean…

The Sine of an angle is the ratio of the length of the leg op…

Theorem 8-1-1

The altitude to the hypotenuse of a right triangle forms two…

Geometric Means (Corollary 8-1-2) (Ign…

The length of the altitude to the hypotenuse of a right trian…

radius (regular polygon)

polygon

remote interior angles

linear scale factor

a line segment that connects the center of a regular polygon…

a two-dimensional closed figure of three or more line segment…

when a triangle has an exterior angles, these angles are the…

the ratio of similarity. the ratio of any pair of correspondi…

radius (regular polygon)

a line segment that connects the center of a regular polygon…

polygon

a two-dimensional closed figure of three or more line segment…

Postulate 1-1 Through any two points,…

Postulate 1-2 If two distinct lines in…

Postulate 1-3 If two distinct _______i…

Postulate 1-4 Through any ________poin…

line

point

planes

three noncollinear

Postulate 1-1 Through any two points,…

line

Postulate 1-2 If two distinct lines in…

point

Theorem 8-1

Corollary 1 to Theorem 8-1

Corollary 2 to Theorem 8-1

Theorem 8-2 Pythagorean Theorem

If the altitude is drawn to the hypotenuse of a right triangl…

When the altitude is drawn to the hypotenuse of a right trian…

When the altitude is drawn to the hypotenuse of a right trian…

In a right triangle, the square of the hypotenuse is equal to…

Theorem 8-1

If the altitude is drawn to the hypotenuse of a right triangl…

Corollary 1 to Theorem 8-1

When the altitude is drawn to the hypotenuse of a right trian…

Parallel lines

Skew lines

Alternate interior angles

Same side interior angles

Coplanar lines that do not intersect

noncoplanar and non parallel lines

nonadjacent interior angles that lie on opposite sides of the…

Interior angles that lie on the same side of the transversal

Parallel lines

Coplanar lines that do not intersect

Skew lines

noncoplanar and non parallel lines

Postulate 3-1 Same-side Interior Angle…

Theorem 3-1 Alternate Interior Angles…

Theorem 3-2 Corresponding Angles Theor…

Theorem 3-3 Alternate Exterior Angles…

supplementary... m∠1 + m∠2 = 180

congruent

corresponding

exterior

Postulate 3-1 Same-side Interior Angle…

supplementary... m∠1 + m∠2 = 180

Theorem 3-1 Alternate Interior Angles…

congruent

Theorem 8.1 - Polygon Interior Angles…

Corollary to Theorem 8.1 - Interior An…

Theorem 8.2 - Polygon Exterior Angles…

Theorem 8.3

The sum of the measures of the interior angles of a convex n-…

The sum of the measures of the interior angles of a quadrilat…

The sum of the measures of the exterior angles of a convex po…

If a quadrilateral is a parallelogram, then its opposite side…

Theorem 8.1 - Polygon Interior Angles…

The sum of the measures of the interior angles of a convex n-…

Corollary to Theorem 8.1 - Interior An…

The sum of the measures of the interior angles of a quadrilat…

Midsegment Theorem

Perpendicular Bisector Theorem

Converse of the Perpendicular Bisector…

Circumcenter Theorem

The segment connecting the midpoints of two sides of a triang…

In a plane, if a point is on the perpendicular bisector of a…

In a plane, if a point is equidistant from the endpoints of a…

The perpendicular bisectors of a triangle intersect at a poin…

Midsegment Theorem

The segment connecting the midpoints of two sides of a triang…

Perpendicular Bisector Theorem

In a plane, if a point is on the perpendicular bisector of a…

Theorem 8.1... If the altitude is drawn t…

Theorem 8.2... The altitude drawn to the…

Theorem 8.3... The altitude drawn to the…

Theorem 8.4... Pythagorean Theorem... In a r…

, then the two triangles formed are similar to the original t…

The length of this altitude is the geometric mean between the…

The length of a leg of this triangle is the geometric mean be…

equal to the square of the length of the hypotenuse.

Theorem 8.1... If the altitude is drawn t…

, then the two triangles formed are similar to the original t…

Theorem 8.2... The altitude drawn to the…

The length of this altitude is the geometric mean between the…

Postulate 1.1

Postulate 1.2

Postulate 1.3

Postulate 1.4

Through any two points, there is exactly one line

If two lines intersect, they intersect in exactly one point

If two planes intersect, they intersect at exactly one line

Through any three noncollinear points, there is exactly one p…

Postulate 1.1

Through any two points, there is exactly one line

Postulate 1.2

If two lines intersect, they intersect in exactly one point

Parallel Postulate

Perpendicular Postulate

Corresponding Angles Postulate

Alternate Interior Angles Theorem

If there is a line and a point not on the line, then there is…

If there is a line and a point not on the line, then there is…

If two parallel lines are cut by a transversal, then the pair…

If two parallel lines are cut by a transversal, then the pair…

Parallel Postulate

If there is a line and a point not on the line, then there is…

Perpendicular Postulate

If there is a line and a point not on the line, then there is…

Theorem 8.1

Theorem 8.2

Theorem 8.3

Theorem 8.4

If the altitude is drawn to the hypotenuse of a right triangl…

The altitude drawn to the hypotenuse of a right triangle sepa…

The altitude drawn to the hypotenuse of a right triangle sepa…

Pythagorean Theorem... In a right triangle, the sum of the squar…

Theorem 8.1

If the altitude is drawn to the hypotenuse of a right triangl…

Theorem 8.2

The altitude drawn to the hypotenuse of a right triangle sepa…

Center of dilation

Contraction

Dilation

Expansion

The point in a dilation through which every line connecting a…

A dilation in which the preimage is reduced in size

A nonrigid transformation that produces an image that is the…

A dilation in which the preimage is enlarged in size

Center of dilation

The point in a dilation through which every line connecting a…

Contraction

A dilation in which the preimage is reduced in size

they are congruent.... Right Angles Cong…

they are congruent.... Congruent Supplem…

they are congruent.... Congruent Supplem…

they are congruent.... Congruent Complem…

If two angles are right angles, then

If two angles are supplementary to the same angle, then

If two angles are supplementary to congruent angles, then

If two angles are complementary to the same angle, then

they are congruent.... Right Angles Cong…

If two angles are right angles, then

they are congruent.... Congruent Supplem…

If two angles are supplementary to the same angle, then

Theorem 8.1

Theorem 8.2

Theorem 8.3

Theorem 8.4

If the altitude is drawn to the hypotenuse of a right triangl…

The altitude drawn to the hypotenuse of a right triangle sepa…

The altitude drawn to the hypotenuse of a right triangle sepa…

Pythagorean Theorem... In a right triangle, the sum of the squar…

Theorem 8.1

If the altitude is drawn to the hypotenuse of a right triangl…

Theorem 8.2

The altitude drawn to the hypotenuse of a right triangle sepa…

Postulate 1-1

Postulate 1-2

Postulate 1-3

Linear Pair Postulate

Through any 2 points there is a line

If 2 lines intersect, they intersect in exactly 1 point

If 2 planes intersect, they intersect in exactly 1 line

If two angles form a linear pair, then they are supplementry

Postulate 1-1

Through any 2 points there is a line

Postulate 1-2

If 2 lines intersect, they intersect in exactly 1 point

Conclusion

Conditional Statement

Conjecture

Contrapositive

In a conditional statement (if-then statement), the statement…

A statement written in an if then form... A statement of the for…

An educated guess based on known information... pg 78

The statement formed by negating both the hypothesis(educated…

Conclusion

In a conditional statement (if-then statement), the statement…

Conditional Statement

A statement written in an if then form... A statement of the for…

Postulate 1-1

Postulate 1-2

Postulate 1-3

Postulate 1-4

There is exactly one line through any two points.

Two distinct intersecting lines intersect at exactly one line.

Two distinct intersecting planes intersect at exactly one line.

There is one plane through any three noncollinear points.

Postulate 1-1

There is exactly one line through any two points.

Postulate 1-2

Two distinct intersecting lines intersect at exactly one line.

8-1: If the altitude is drawn to the h…

Corollary 1: When the altitude is draw…

Corollary 2: When the altitude is draw…

8-2: (Pythagorean Theorem) In a right…

then the two triangles formed are similar to the original tri…

the length of the altitude is the geometric mean between the…

each leg is the geometric mean between the hypotenuse and the…

the square of the hypotenuse is equal to the sum of the squar…

8-1: If the altitude is drawn to the h…

then the two triangles formed are similar to the original tri…

Corollary 1: When the altitude is draw…

the length of the altitude is the geometric mean between the…

Postulate 2.1

Postulate 2.2

Postulate 2.3

Postulate 2.4

Through any two points, there is exactly one line.

Through any three noncollinear points, there is exactly one p…

A line contains at least two points.

A plane contains at least three noncollinear points.

Postulate 2.1

Through any two points, there is exactly one line.

Postulate 2.2

Through any three noncollinear points, there is exactly one p…

Postulate 1-1

Postulate 1-2

Postulate 1-3

Postulate 1-4

Through any two points there is exactly one line.

If two lines intersect, then they intersect in exactly one po…

If two planes intersect, they intersect in exactly one plane.

Through any three non collinear points there is exactly one p…

Postulate 1-1

Through any two points there is exactly one line.

Postulate 1-2

If two lines intersect, then they intersect in exactly one po…

Ruler Postulate

Segment Addition Postulate

Congruent Figures

Congruent Segments

1. The points on a line can be paired with the real numbers i…

If B is between A and C, then AB+BC=AC

Have the same size and shape

Have equal lengths

Ruler Postulate

1. The points on a line can be paired with the real numbers i…

Segment Addition Postulate

If B is between A and C, then AB+BC=AC

through any two points

if two distinct lines intersect

if two distince planes intersect

through any three noncollinear points

there is exactly one line

then they intersect in exactly one point

then they intersect in exactly one line

there is exactly one plane

through any two points

there is exactly one line

if two distinct lines intersect

then they intersect in exactly one point

point

line

plane

collinear

a location; has no length, no width, no thickness

extends in one dimension; has length but no width and no thic…

A flat surface that has length and width but no thickness

Points that lie on the same line.

point

a location; has no length, no width, no thickness

line

extends in one dimension; has length but no width and no thic…

Pythagorean Theorem

Pythagorean Triple

Obtuse Triangle

Acute Triangle

A triangle is a right if the sum of the squares of the length…

a set of nonzero whole numbers a, b, and c that make the equa…

The square of the length of the longest side of a triangle is…

The square of the length of the longest side of a triangle is…

Pythagorean Theorem

A triangle is a right if the sum of the squares of the length…

Pythagorean Triple

a set of nonzero whole numbers a, b, and c that make the equa…

Perimeters of Similar Polygons Theorem

Angle-Angle (AA) Similarity Postulate

Side-Side-Side (SSS) Similarity Theorem

Side-Angle-Side (SAS) Similarity Theorem

If two polygons are similar, then the ratio of their perimete…

If two angles of one triangle are congruent to two angles of…

If the lengths of the corresponding sides of two triangles ar…

If an angle of one triangle is congruent to an angle of a sec…

Perimeters of Similar Polygons Theorem

If two polygons are similar, then the ratio of their perimete…

Angle-Angle (AA) Similarity Postulate

If two angles of one triangle are congruent to two angles of…