Chapter 12

Question 1

def. similar

Answer 1

given a corr. between two triangles, if corresponding angles are congruent and corresponding sides are proportional, then the correspondence is called a similarity.
the triangles are similar

Question 2

Basic Proportionality Thm. (BPT)

Answer 2

If a line is parallel to one side of a triangle and it intersects the other two sides, it will cut off segments that are proportional to the two sides of the triangle.

Question 3

Converse of the BPT

Answer 3

If a line intersects two sides of a triangle and cuts off segments that are proportional to the two sides of the triangle, the line is parallel to the third side.

Question 4

Angle Bisector Proportionality Thm.

Answer 4

The bisector of an angle of a triangle separates the opposite side into two segments which are proportional to their adjacent sides of the triangle.

Question 5

Parallels-Proportional Segments Thm.

Answer 5

If 3+ lines are each cut by two transversals, then the intercepted segments on the two transversals are proportional.

Question 6

AAA~ Thm.

Answer 6

Given a correspondence btwn 2 triangles, if 3 angles of triangle one are congruent to the corresponding angles of triangle two, then the two triangles are similar.

Question 7

AA~ corollary

Answer 7

As a consequence of the triangle angle sum thm., if two angles of triangle one are congruent to the corresponding angles of triangle two, then the two triangles are similar.

Question 8

SAS~ Thm.

Answer 8

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

Question 9

SSS~ Thm.

Answer 9

If all three sides of one triangle are proportional to the corresponding sides of the second triangle, then the two triangles are similar.

Question 10

Theorem 1

Answer 10

given a right triangle, the altitude to the hypotenuse separates the triangle into two smaller triangles, which are similar to each other and to the original triangle.

Question 11

Theorem 2

Answer 11

given a right triangle and the altitude to the hypotenuse:

a. the altitude is the geometric mean of the two segments into which the hypotenuse has been separated.
b. each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg

Question 12

finding geometric mean

Answer 12

x/g = g/y

cross multiply

Question 13

finding arithmetic mean

Answer 13

x + y/2