Geometry Chapter 7 (Similarity)

Angle-Angle (AA) similarity postulate

If two angles if one triangle are congruent to two angles of another triangle, then the triangles are similar

Side-Angle-Side (SAS) similarity theorem

If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar

Transitive property of similarity

If 🔺ABC is similar to 🔺DEF and 🔺DEF is similar to 🔺XYZ, then 🔺ABC is similar to 🔺XYZ

Symmetric property of similarity

If 🔺ABC is similar to 🔺DEF, then 🔺DEF is similar to 🔺ABC

Side-Side-Side (SSS) similarity theorem

Of the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar

Triangle Angle Bisector Theorem

an angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides

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 a side of a triangle intersects the other two sides, then it divides those sides proportionally

Proportional Perimeters and Areas Theorem

if the similarity ratio of two similar fingures is a/b, then the ratio of their perimeters is a/b, and the ratio of their areas is a squared/ b squared, or (a/b)squared

Two Transversal Proportionality

if three or more parallel lines intersect two transversals, then they divide the transversals proportionally

Cross Products Property

in a proportion, if a/b = c/d and b and d don't = 0, then ad = bc

Property of Proportions

the proportion a/b = c/d is equivalent to the following: ad = bc, b/a = d/c, a/c = b/d ex: the proportion 1/3 = 2/6 is equivalent to the following: 1(6) = 3(2), 3/1 = 6/2, 1/2 = 3/6

Similar Polygons

two polygons that have congruent corresponding angles and proportional corresponding side lengths

Reflexive property of similarity

🔺ABC is similar to 🔺ABC

Similarity Ratio

🔺ABC is similar to 🔺DEF 🔺ABC has side lengths: 3, 4, and 5 🔺DEF had side lengths: 6, 8, and 10 AB/DE = AC/DF = BC/EF = 1/2

Area Ratio

🔺ABC is similar to 🔺DEF 🔺ABC has side lengths: 3, 4, and 5 🔺DEF had side lengths: 6, 8, and 10 area 🔺ABC / area 🔺DEF = 6/24 = 1/4 = (1/2)squared

Perimeter Ratio

🔺ABC is similar to 🔺DEF 🔺ABC has side lengths: 3, 4, and 5 🔺DEF had side lengths: 6, 8, and 10 perimeter 🔺ABC/ perimeter 🔺DEF = 12/24 = 1/2