Geometry Chapter 5+6 Flash Cards

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Side-Angle-Side Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If angle A is congruent to angle D and BA/ED=AC/DF, then triangle BAC is similar to triangle EDF.

Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent. The lines containing AF, BE, and CD meet at G.

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The medians of triangle ABC meet at P and AP=2/3 AE, BP=2/3 BF, and CP=2/3 CD.

Concurrency of Perpendicular Bisectors of a Triangle Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. If PD, PE, and PF are perpendicular bisectors, then PA=PB=PC.

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

Dilations and Similarity

If a dilation can be used to move one figure onto another, the two figures are similar. Triangle ABC is similar to triangle A'B'C'

Combining Dilations and Rigid Motions

If a dilation followed by any combination of rigid motions can be used to move one figure onto the other, the two figures are similar.

Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If EF is parallel to BC, then AE/ED=AF/FC.

Converse of the Angle Bisector Theorem

If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If PA is perpendicular to CA and PB is perpendicular to CB and PA=PB, then CP bisects angle ACB.

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If CP bisects angle ACB and PA is perpendicular to CA and PB is perpendicular to CB, then PA=PB.

Side-Side-Side (SSS) Similarity Theorem

If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If AB/RS=BC/ST=CA/TR, then triangle ABC is similar to triangle RST.

Angle-Angle (AA) Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Triangle CAB is similar to triangle HFG.

Corresponding Lengths in Similar Polygons

If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

Perimeters of Similar Polygons

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

Converse of the Perpendicular Bisector Theorem

In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA=DB, then D lies on the perpendicular bisector of AB.

Perpendicular Bisector Theorem

In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA=CB.

How to Write an Indirect Proof (Proof by Contradiction)

Step 1: Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true. Step 2: Reason logically until you reach a contradiction. Step 3: Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false.

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

TS/SR=ZY/YX

Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. If AP, BP, and CP are angle bisectors of triangle ABC, then PD=PE=PF.

Coordinate Notation for a Dilation

You can describe a dilation with respect to the origin with the notation (x,y) -> (kx, ky) where k is the scale factor. If 0 < k < 1, the dilation is a reduction. If k > 1, the dilation is an enlargement.

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

the measure of angle B > the measure of angle C, so AC > BA.

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

AC>AB, so the measure of angle B > the measure of angle C.

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

BD/DC=AB/AC.