The document discusses similarity of triangles and geometric figures, noting that two figures are similar if they have the same shape but not necessarily the same size, and outlining the AA, SAS, SSS, and right triangle similarity theorems. It also reviews ratios, proportions, and how proportions can be used to show that corresponding segments of similar figures are proportional. The learning objectives are to illustrate similarity of figures and prove the conditions for similarity of triangles using these various theorems.

1. SIMILARITY OF TRIANGLES
Guided Learning Activity Kit in Mathematics
Grade 9 Quarter 3 Week 6&7
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2. Introduction:
SIMILAR
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3. Alternation Property
Inverse Property
Addition Property
Subtraction
Property
Sum Property of the
Original Proportion
Solving distance across the river and
the height of the skyscraper.
Scale drawing of maps and floor
plans
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4. Learning Competency
The learner illustrates similarity of figures (M9GE-IIIg-1); and
 proves the conditions for similarity of triangles.
1.1 SAS similarity theorem
1.2 SSS similarity theorem
1.3 AA similarity theorem
1.4 right triangle similarity theorem
1.5 special right triangle theorems.
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5. Objectives:
At the end of this guided learning activity kit, you are expected to:
1. illustrate similarity of figures; and
2. prove the conditions for similarity of triangles
a. AA Similarity Theorem
b. SAS Similarity Theorem
c. SSS Similarity Theorem
d. Right Triangle Similarity Theorem
e. 45-45-90 Right Triangle Theorem
f. 30-60-90 Right Triangle Theorem
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6. Review:
Recall that a ratio is an expression of the relationship of two numbers. Given two real
numbers a and b, where b ≠ 0, a ration of a to b is the quotient a divided by b. May be written
as:
a to b a:b
𝑎
𝑏
a/b
Proportion is an equation of two or more ratios. If the ratio a:b and c:d are equal, then we
write a:b = c:d or
𝑎
𝑏
=
𝑐
𝑑
(where b and d are not equal to zero). The symbol = is read “is
proportional to”.

7. Illustrative example:
Examples:
Find the value of x in each proportion.
1. 2 : 4 = 8 : x 2.
3
𝑥
=
24
40
Solution:
1. 2(x) = 4(8) 2. 3(10) = 24(x)
2x = 32 120 = 24x
x = 16 5 = x

8. If a, b, and c are positive real numbers, and
𝑎
𝑏
=
𝑏
𝑑
, then b is called the
geometric means of a and c. The proportion suggest that b2 = ac by the
theorem above, and hence, b = 𝑎𝑐.
Examples:
Find the geometric means of:
1. 4 and 9 2. 7 and 14
Solution:
1. The geometric mean b of 4 and 9 is
b = (4)(9) = 36 = 6
2. The geometric mean b of 7 and 14 is
b = (7)(14) = 98 = 7 2

9. And proportion also involves geometric figures like line segments
which is called proportional segments. Corresponding segments
are proportional if the segments of one figure have the same ratio
as the segments of the other.
In the above figure, MO corresponds to XW, MN corresponds to XY,
and ON corresponds to WY. Segments MN and XY are said to be
divided proportionally.

10. Discussion:
Two geometric figures are similar if they have the same shape. They may or may
not have the same size. For example, any two or more line segments are similar,
or any two or more circles are similar, or any two or more squares are similar.
Illustrations:

11. Squares are similar because their corresponding angles are congruent, and
the ratio of their corresponding vertices are proportional when paired.
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12. Self-Similarity
Questions:
1. How many self-similar
hexagons are there?
2. Do you know how this
figure is formed? Study the
initial steps on how the
hexagon is replicated many
times in decreasing sizes.
Describe each step in words.
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