A Postulate for Similar Triangles Lesson 7.4

Pre-AP Geometry

Lesson Focus The focus of this lesson is a postulate for establishing when two triangles are similar. The postulate states that two triangles are similar whenever two pairs of angles are congruent.

Similar Triangles In general, to prove that two polygons are similar, you must show that all pairs of corresponding angles are equal and that all ratios of pairs of corresponding sides are equal. In triangles, though, this is not necessary.

AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

A B D E C

AA Similarity Postulate Showing  ABC  DEF: In  ABC, m  A + m  B + m  C = 180  m  A + 100 m  A = 60   + 20  = 180  But in  DEF, m  D = 60  So, m  A = m  D (continued)

AA Similarity Postulate Additionally, because the triangles are now similar, m  C = m  F and, .

DE EF



DF

AA Similarity Postulate In the figure, m  1 = m  2, because vertical angles are equal.

Also, m

 R = m  T and m 

Q = m

 U, because if two parallel lines are cut by a transversal, then the alternate interior angles are equal. So by the AA Similarity Postulate, 

QRS

  UTS.

AA Similarity Postulate In  MNO, MN = NO, and in  PQR, PQ = QR; m  M = m  O and m  P = m 

R.

(If two sides of a triangle are equal, the angles opposite these sides have equal measures.) Also, in  MNO, m Because m   M = m M + m  N + m  O and m   O = 180° and in  PQR, m P = m 

R

 P + m  Q + m  R = 180°. So, m  M = m  P, and m Therefore, 

MNO

∼  O = m 

R.

 PQR ( AA Similarity Postulate).

AA Similarity Postulate

m

 C = m 

m

 A = m  F (All right angles are equal.) D (They are indicated as equal in the figure.) 

ABC

  DEF ( AA Similarity Postulate)

Historical Note The earliest surviving Chinese book on mathematics and astronomy dates from around 2200 years ago. Along with presenting a theorem equivalent to the Pythagorean theorem, it describes how to use similar right triangles to survey heights, depths, and distances.

Written Exercises Problem Set 7.4A, p.257: # 2 - 12 (even)

Written Exercises Problem Set 7.4B, Handout 7-4