A Postulate for Similar Triangles

1
A Postulate for Similar Triangles

• Lesson 7.4
• Pre-AP Geometry

2
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.

3
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.

4
AA Similarity Postulate

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

5
AA Similarity Postulate

• Showing ?ABC ??DEF
• In ?ABC, m?A m?B m?C 180?
• m?A 100? 20? 180?
• m?A 60?
• But in ?DEF, m?D 60?
• So, m?A m?D (continued)

6
AA Similarity Postulate

• Additionally, because the triangles are now
  similar,
• m?C m?F and, .

7
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.

8
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 ?M m ?N m ?O 180 and in
  ?PQR, m ?P m ?Q m ?R 180.
• Because m ?M m ?O and m ? P m ?R
• So, m?M m?P, and m?O m ?R.
• Therefore, ?MNO ?PQR ( AA Similarity
  Postulate).

9
AA Similarity Postulate

• m?C m? F (All right angles are equal.)
• m? A m? D (They are indicated as equal in the
  figure.)
• ? ABC ? ? DEF ( AA Similarity Postulate)

10
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.

11
Written Exercises

• Problem Set 7.4A, p.257 2 - 12 (even)

12
Written Exercises

• Problem Set 7.4B, Handout 7-4