# A Postulate for Similar Triangles

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```					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

D               E

B                        C
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)
AA Similarity Postulate
Additionally, because the triangles are now similar,
mC = mF and, AB  BC  AC .
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 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).
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)
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

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