10 Applied Maths Targeted Teaching - Takeaway
Similarity and Congruence
Name ______________________
Advisory ______
TT Teacher________
T2 Teaching Focus
Similar Triangles are ones how have the same angles and sides that are in ratio to each other.
There are three ways to determine if triangles are similar.
AA – Two angles are equal. Since all triangles have angles adding up to 180° the other angle
must be the same.
SAS - If two triangles have two pairs of sides in the same ratio and the included angles are also
equal, then the triangles are similar.
SSS - If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
Other shapes can also be similar if their sides are in ratio and the angles equal. All squares are
technically similar.
10 Applied Maths – Similarity and Congruence
Folland 2015
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Congruent Triangles are a special type of similar triangle that have the same angles and side
lengths. They are shapes that have been rotation, reflection or repositioning.
SSS - If three sides of one triangle are equal to three sides of another triangle, the triangles are
congruent.
SAS - If two sides and the included angle of one triangle are equal to the
corresponding sides and angle of another triangle, the triangles are congruent.
ASA - If two angles and the included side of one triangle are equal to the
corresponding angles and side of another triangle, the triangles are congruent.
AAS - If two angles and the non-included side of one triangle are equal to the
corresponding angles and side of another triangle, the triangles are congruent.
We will work more with similar triangles when we examine trigonometry.
(Images sourced from mathsisfun.com)
10 Applied Maths – Similarity and Congruence
Folland 2015
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T2 Teaching Example
Find lengths a and b in the triangles below.
1. Determine if triangles are similar or congruent (or neither)
a. All three angles in the triangle are equal (represented by the singe, double and triple
curves)
b. The known sides are not the same. So the triangles must be similar due to the AAA
rule.
2. If similar determine the ratio between sides
a. The corresponding sides opposite the second angle (two curves) have sizes of 6.4
and 8 units.
b. The enlargement ratio for this triangle is 6.4/8
3. Using the ratio determine the length of the unknown sides
a. The ratio of the known pair is equal to the unknown and its matching side
b. Using the sides opposite the single angle
𝑎
7
=
6.4
8
𝑟𝑒𝑎𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑔𝑖𝑣𝑒𝑠 𝑎 = 7 ×
6.4
8
= 5.6
c. Using the sides opposite the triple angle
𝑏
6.4
6.4
=
𝑟𝑒𝑎𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑔𝑖𝑣𝑒𝑠 𝑏 = 6 × = 4.8
6
8
8
Touch Base Tasks:
Given that the two triangles below ABC and DEF are similar triangles find the unknown length of
X. (show all working):
A
D
5cm
α
α
X cm
θ
B
C
4cm
F
θ
12cm
10 Applied Maths – Similarity and Congruence
E
Folland 2015
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Identify the four pairs of congruent triangles. State which congruence rule was used to establish
their congruence (not all diagrams to scale)
5cm
5cm
2cm
A
2cm
B
2cm
80
2cm
C
60
20
5cm
G
60
5cm
5cm
6cm
D
5cm
F
E
80
5cm
2cm
H
60
80
I
J
2cm
60
80
5cm
5cm
5cm
3cm
4cm
5cm
L
K
4cm
Pair
10 Applied Maths – Similarity and Congruence
Reason
Folland 2015
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3cm
10 Applied Maths – Similarity and Congruence
Folland 2015
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