- Hales, a Greek mathematician, was the first to measure the height of a pyramid using similar triangles. He showed that the ratio of the height of the pyramid to the height of the worker was the same as the ratio of the heights of their respective shadows. - The document discusses using similar triangles to solve problems involving finding unknown lengths, such as measuring the height of a pyramid based on the shadow lengths of the pyramid and a worker of known height. - Examples are provided of determining if two triangles are similar based on proportional sides or equal corresponding angles, and using similarities between triangles to find unknown lengths.

1. Mathematics for Junior High School Grade 9 / 9
11..22
hales, a Greek mathematician, was the first
to measure the height of a pyramid using
geometrical principles. He showed that the ratio
of the height of the pyramid to the height of the
worker was the same as the ratio of the heights of
their respective shadows.
Can you find the height of the pyramid? (You are supposed to
solve this problem in Task 1.2).
Similar triangles can help you solve such a problem. How can you
identify two similar triangles?
In the previous lesson you learned how to determine the similarity
of two polygons. Now you are learning how to determine whether
two triangles are similar.
TT
What are you going to
Properties of Two Similar Triangleslearn?
to determine
requirements for two
similar triangles
to determine the ratio of
the sides of two triangles
to find the side lengths
to solve a problem
involving similarity
concept
Key terms
History
:
• similar triangles
• corresponding sides
Figure 1.5
Height of
pyramid
AB = ?
Shadow of pyramid BC = 576
feet
Shadow of of a worker
FD = 6 feet
Height of worker
EF = 5 feet

2. Using a ruler and a protractor,
• draw ΔDEF with ∠D = 35°, ∠ F = 80°, and DF = 4cm
• draw ΔTRS with ∠T = 35°, ∠ S = 80°, and ST = 7cm
• measure the lengths of EF, ED, RS, and RT
• determine the ratios of , and
FD EF ED
ST RS RT
.
Write down the results on the table below.
Δ DEF Δ TRS Values of Ratios
EF ED RS RT ST
FD
RS
EF
RT
ED
Are ΔDEF and ΔTRS similar?
Given two triangles, if the corresponding angles are
the same measure, then the two triangles are similar.
Using a ruler and a protractor,
• draw triangle ABC with AB = 8 cm, BC = 6 cm, and AC = 7
cm.
• draw triangle PQR with PQ = 4 cm, QR = 3 cm, and PR =
3.5 cm.
• measure ∠ A, ∠ B, ∠ C, ∠ P, ∠ Q, ∠ R
• determine whether ∠ A = ∠ P, ∠ B = ∠ Q , ∠ C = ∠ R
Are ΔABC and ΔPQR similar?
Given two triangles, if the corresponding sides are
proportional, then the two triangles are similar.
10 / Student’s Book – Similarity and Congruency

3. Check whether ΔPQR and ΔMNO are similar. What about the
corresponding angles?
Mathematics for Junior High School Grade 9 / 11
a Find whether ΔUTV and ΔUSR below are similar.
b Write down the proportions of the corresponding sides.
Answer:
3
1
45
15
==
MO
PR
3
1
21
7
==
ON
RQ
3
1
30
10
==
MN
PQ
3
1
===
ON
RQ
MN
PQ
MO
PR
So ΔPQR and ΔMNO are similar.
Consequently, ∠ R = ∠ O, ∠ P = ∠ M, and ∠ Q = ∠ N
15 45
N
30
21Q
P
107
R MO
>
>
U
V T
R S

4. In the figure on the right,
AB // DE
a Show that ΔABC and
ΔEDC are similar.
b Write down the
proportions of the
corresponding sides.
The Length of an Unknown Side of Two Similar
Triangles
Look at the figure on the left.
BC // DE. You have found that ΔADE
and ΔABC are similar. The length of
AD = p and DB = q.
Since ΔADE and ΔABC are similar,
.
.
AD AE
AB AC
p x
p q x
=
=
y+ +
p(x + y) = x(p + q)
px + py = px + qx
py = qx
y
x
q
p
=
Therefore, the proportions of the segments of the two legs of
triangle ABC are:
A
p x
D E
y
q
B C
y
x
q
p
=
C
A B
ED
12 / Student’s Book – Similarity and Congruency

5. Given a triangle, if there is a line parallel to one of the sides
of the triangle, then the line divides the other two sides into
two proportional segments.
In the figure on the
right
A
C
E F
D
BH
G
I
>
>
>
EFCDAB ////
Complete the following
statements:
a.
.....
CE
BD
AC
=
b.
HIIE
CE .....
=
c.
GFGE
GH .....
=
Look at the figure on the right, DE // AB
A B
C
D E
x
2
3
3
y
10
a. Show that ΔABC and
ΔDEC are similar.
b. Find the values of x and y.
Mathematics for Junior High School Grade 9 / 13

6. 1. Look at the figures on the right.
R W
40°
70°
P Q
U V
a. Show that ΔPQR and ΔUVW
are similar.
b. Find the pairs of the
proportional corresponding
sides.
2. Look at the figures on the left.
a. Show that ΔABC and ΔEFD are
similar.
b. Find the pairs of angles that are
the same measure.
3. Find pairs of similar triangles
from the figure on the right. Give
reasons why the triangles in each
pair are similar.
(Hint: Put in order the angles of
the same measure in the similar
triangles)
Find the values of a, b, c, and d in Figures 5–9 below.
4.
C
D
E
F
B
A
9
15
12
8
10
6
4 cm
8 cm
5 cm
a cm
14 / Student’s Book – Similarity and Congruency

7. d mm
5. 6.
3 mm
16 mm
12 mm
20 m
8 m 25 m
c m
7. 8.
Mathematics for Junior High School Grade 9 / 15
9. A, B, and C are the medians of
sides DF, DE, and FE
a. Given BC = 11, AC = 13, and
AB = 15, find the perimeter of
ΔDEF
b. Given DE = 18, DA = 10, and FC
= 7, find AB, BC and AC.
10. Look at the figure on the right.
Given ∠B = 900,
a. show that ΔADB and ΔABC
are similar and that c2 = pb
b. show that ΔBDC and ΔABC
are similar and that a2 = qb
D
A F
B C
E
C
B
a
c
t
A
p q
D
b
12 cm c cm
8 cm 15 cm
d cm
3 cm
4 cm
a cm
6 cm
10¼

8. 11. If in Figure 1.5 (on page 8) BC = 576
feet and FD = 6 feet (remember that 1
foot = 30.48 cm), find the height of the
pyramid (in feet) based on similarity
concept.
12. In a flag-raising ceremony, your
shadow is 200 cm in length, and
the shadow of the flag pole is 700
cm. If you are 160 cm tall, find the
height of the flag pole.
13.
Given ED // AB, AB = 10, BC = 6, AC = 8,
CD = 5 and GE = 3, find EC, GC, and EF.
A
E
C
D
B
F
G
16 / Student’s Book – Similarity and Congruency