Download Scholarship Geometry Notes 7-3 Triangle Similarity Recall the

Scholarship Geometry Notes
7-3 Triangle Similarity
Recall the reasons for triangle congruence. What were they?
SSS, SAS, ASA, AAS, HL
There are three reasons we can use to prove the triangles are similar.
1) Angle-Angle (AA) Similarity
If two angles of one triangle are congruent to two angles of a second triangle, the triangles are
similar.
2) Side-Side-Side (SSS) Similarity
If the three sides of one triangle are proportional to the three sides of a second triangle, the
triangles are similar.
5
2
2 4 5
= =
4 8 10
10
4
4
8
3) Side-Angle-Side (SAS) Similarity
If two sides of one triangle are proportional to two sides of another, and the included angles are
congruent, then the triangles are similar.
2 4
=
4 8
2
4
4
8
Caution: These rules ONLY work for triangles. For all other polygons, you must show
ALL angles congruent and ALL sides proportional.
Ex. 1: Verify that the triangles are similar and write the similarity statement.
3
3 2
=
=
4.5 4.5 3
Similar by SSS. ΔRPQ ~ ΔSUT
2 5.8
=
1 2.9
Similar by SAS. ΔDEF ~ ΔHJK
m∠B = 180 − 40 − 80 = 60°
E
B
60°
Similar by AA. ΔABC ~ ΔDEF
D
40°
A
80°
C
80°
F
Ex. 2: Explain why the triangles are similar and then find x.
The triangles share angle A, and they both have a 90° angle.
So, they're similar by AA.
AC = 9 + 3 = 12
9 5
=
12 x
9x = 60
60
x=
= 6. 6
9
Ex. 3: Explain why the triangles are similar and find BE and CD.
The triangles share angle A, and ∠ABE ≅ ∠ACD.
So, they're similar by AA.
AC = 3 + 4.5 = 7.5
3
7.5
=
x x+6
3( x + 6) = 7.5x
3x + 18 = 7.5x
18 = 4.5x
18
=4
4.5
BE = 4 CD = 4 + 6 = 10
x=
Ex. 4: Find BA.
AF = 17 BF = x
x
6.5
=
17 + x 24
24 x = 6.5(17 + x )
AB = 17 + x
24 x = 110.5 + 6.5x
17.5x = 110.5
110.5
x=
≈ 6.31
17.5
BA = 17 + 6.31 = 23.31 ft