CH 12 Honors Geometry Theorems

Reflexive Property

any geometric figure is similar to itself

Symmetric property

A similarity between two geometric figures may be expressed in either order

Transitive property

Two geometric figures similar to the same geometric figure are similar to each other

for any given triangle there exists a….

similar triangle with any given ratio of similitude

In a proportion the product of the means is equal to…

the product of the extremes

In a proportions, the means may be

interchanged

If the product of two pairs of factors are equal

the factors of one pair can be the means and the factors of the other extremes of a proportion

A line segment joining the midpoints of two sides of a triangle is parallel to..

the third side and its length is one-half the length of the third side

Two line segments are divided proportionally if and only if..

the ratio of the length of a part of one segment to the length of the whole is equal to the ratio of the corresponding lengths of the other segment

Two triangles are similar two angles of one of the triangle are…

congruent to two corresponding angles of the other (AA~)

Two triangles are similar if the three ratios..

of corresponding sides are equal (SSS~)

Two triangles are similar is the ratios of two pairs of corresponding sides are..

equal and the corresponding angles included between these sides are congruent (SAS~)

A line is parallel to one side of a triangle and intersects the other two sides if and only if..

the points of intersection divide the sides proportionally

Under a dilation, angle measure is

preserved

Under a dilation, midpoint is

preserved

Under a dilation, Collinearity is..

preserved

if two triangles are similar, the lengths of corresponding altitudes have..

the same ratio as the length of any two corresponding sides

If Two triangles are similar, the lengths of corresponding medians have…

the same ratio as the lengths of any two corresponding sides

If two triangles are similar, the lengths of corresponding angle bisectors have…

the same ratio as the lengths of any two corresponding sides

Any two medians of a triangle intersect in a point that divides each median into the ratio..

2:1

The medians of a triangle are

concurrent

The altitude to the hypotenuse of a right triangle..

divides the triangle into two triangles that are similar to each other and to the original triangle

The length of each leg of a right triangle is..

the mean proportional between the length of the projection of that leg on the hypotenuse and the length of the hypotenuse

The length of the altitude to the hypotenuse of a right triangle is the…

mean proportional between the lengths of the projections of the legs on the hypotenuse

A triangle is a right triangle if and only if..

the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides