The document outlines 9 theorems regarding similar triangles, including: (1) two triangles with two congruent angles are similar; (2) two triangles with a congruent angle between proportional sides are similar; and (3) two triangles with three proportional sides are similar. It also provides examples of applying the theorems to find missing side lengths in similar triangles. The document concludes with practice problems for students.

1. Theorem on Similarity
Minimum Conditions
for Similar Triangles

2. Theorem 1: AA
• Two triangles with 2 corresponding
congruent angles are similar.

3. Theorem 2: SAS
• Two triangles with a congruent angle
contained between corresponding sides of
proportional length are similar.

4. Theorem 3: SSS
• Two triangles whose three corresponding
sides are proportional are similar.

5. Theorem 4:
• Any line secant to 2 sides of a triangle and
parallel to the 3rd side forms a triangle
similar to the first.

6. Theorem 5
• Any line secant to 2 sides of a triangle and
parallel to the 3rd side divides the 2 sides
into segments whose lengths are
proportional.

7. Theorem 6
• Parallel lines that intersect secants
produce segments whose lengths are
proportional.

8. Theorem 7
• Any line segment joining the midpoints of
2 sides of a triangle is parallel to the 3rd
side, and its length is ½ the length of the
3rd side.

9. Activities
• P. 401, Q. 5,6,7,9,10

10. Theorem 8
• The perimeters of similar figures are in the
same ratio as the measures of the
corresponding sides.

11. Theorem 9
• The ratio of the areas of similar figures is
equal to the square of the ratio of the
measures of the corresponding sides.
• k = ratio of lengths or perimeters
• k2 = ratio of areas
• k3 = ratio of volumes

12. Exam Question
In the figure to the right, segments DE and CB
are parallel and they measure 6 units and
10 units respectively. Segment EB measures
3 units.
B
A
C
D
E
6
10
?
3
What is the measure of segment AE?
A) 2 units C) 5 units
B) 4.5 units D) 7.5 units

13. Exam Question
•
In the following figure, right triangles ABC and CDE are similar. Angle ABC measures 60;
segments AB and CE measure 4 cm.
What is the measure of segment DE?
A
B
C D
E
A) 2 cm C) 6 cm
B) 4 cm D) 8 cm

14. Exam Question
In the diagram below, triangle ABC has a right angle at C and measurements as indicated.
D is the midpoint of AB .
A
B
C
E
D
80
48
64
What is the measure of segment EC?
A) 32.0 units C) 14.0 units
B) 24.0 units D) 10.7 units

15. Activity
• P. 409, Q. 1,2,3,13,14,15
• P. 420, Q. 1-6