Download 1. SSS (side, side, side) 2. SAS (side, angle, side)

MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
CONGRUENT TRIANGLES
If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent
ROTATION / TURN
REFLECTION / FLIP
TRANSLATION/SLIDE
After any of those transformation (turn, flip or slide), the shape still has the same size, area, angles,
and line lengths.
How to Find if Triangles are Congruent
Two triangles are congruent if they have:

Exactly the same three sides and exactly the same three angles.
But we don’t have to know all three sides and all three angles…usually three out of the six is enough.
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS, and HL.
1. SSS (side, side, side)
SSS stands for "side, side, side" and means
For example:
is
congruent
to:
that we have two
triangles with all three
sides equal.
2. SAS (side, angle, side)
SAS stands for "side, angle, side" and means
that we have two
triangles where we know
For example:
is congruent
to:
two sides and the
included angle are equal.
Page 1 of 6
MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
3. ASA (angle, side, angle)
ASA stands for "angle, side, angle" and
means that we
have two triangles
where we know
two angles and the
included side are
equal.
For example:
is congruent
to:
4. AAS (angle, angle, side)
AAS stands for "angle,
angle, side" and means that
we have two triangles where
we know two angles and the
non-included side are equal.
For example:
is congruent
to:
5. HL (hypotenuse, leg)
HL stands for “Hypotenuse, Leg. This one applies only to right angled-triangles!
is congruent to
MARKING
When two triangles are congruent we often mark corresponding sides and angles like this:
ΔABC  ΔRST
Page 2 of 6
MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
D
EXAMPLE #1:
Are these triangles congruent? Which congruence
6cm
A
sufficiency condition applies?
6 cm
B
a) State the congruency statement.
i.e. Δ ___ ___ ___  Δ ___ ___ ___
12 cm
E
12 cm
8 cm
8 cm
F
C
EXAMPLE #2:
ΔNPQ  ΔRST. State the values of x, y, and z.
EXAMPLE #3:
K
ΔEFG  ΔHJK. State the values of
E
x, y, and z.
80
8.5 cm
6 cm
F
z
x
H
G
9.6 cm
y
62
EXAMPLE #4:
J
Show that these triangles are congruent. State the congruency statement, the sufficiency
condition, and state all evidence.
8 cm
y
A
B
a) Determine the values of x, y, and z.
5.2 cm
z
E
100
b) Determine the values of x, y, and z.
x
40
D
C
Page 3 of 6
MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
SIMILAR TRIANGLES
Similar triangles have:
 all their angles equal
 corresponding sides have the same ratio
These triangles are SIMILAR.
(Equal angles have been marked with the same number of arcs)
If one shape can become another using Resizing (dilation, contraction, compression, enlargement) then
these Shapes are Similar!
How to Find if Triangles are Similar
But we don't need to know all three sides and all three angles ...two or three out of the six is usually
enough. There are three ways to find if two triangles are similar: AA, SAS and SSS:
AA
AA stands for "angle, angle" and means that the
triangles have two of their angles equal.
If two triangles have two of their angles equal, the
triangles are similar.
Example:
SAS
SAS stands for "side, angle, side" and means
that we have two triangles where:
 the ratio between two sides is the same as
the ratio between another two sides
 and we we also know the included angles are
equal.
Example
In this example we can see that:
 one pair of sides is in the ratio of 21 : 14 = 3 : 2
 another pair of sides is in the ratio of 15 : 10 = 3 : 2
 there is a matching angle of 75° in between them
If two triangles have two pairs of sides in the
same ratio and the included angles are also
equal, then the triangles are similar.
Page 4 of 6
MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
SSS
SSS stands for "side, side, side" and means that
we have two triangles with all three pairs of
corresponding sides in the same ratio.
Example:
If two triangles have three pairs of sides in the
same ratio, then the triangles are similar.
CORRESPONDING SIDES
In similar triangles, the sides facing the equal angles are always in the same ratio.
For example:
Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.
What are the corresponding lengths?

The lengths 7 and a are corresponding (they face the angle marked with one arc)

The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs)

The lengths 6 and b are corresponding (they face the angle marked with three arcs)
CALCULATING THE LENGTHS OF CORRESPONDING SIDES
It may be possible to calculate lengths we don't know yet. We need to:
Step 1: Find the ratio of corresponding sides in pairs of similar triangles.
Step 2: Use that ratio to find the unknown lengths.
Ex1: Find lengths a and b of Triangle S.
Page 5 of 6
MPM2D1
Day 2: Similar vs. Congruent Triangles
Date: ____________
Chapter 7: Trigonometry of Right Triangles
Example 1:
a. Explain why ΔABC ~ ΔDEF. Justify your answer.
F
B
x
A
9
a
5
o
e
7
C
E
x
b. Determine the values of a and e.
10
o
D
Example 2: Are the two triangles in the diagram similar? Explain your reasoning.
Page 334 # 7, 8
Page 348 # 5, 6, 7, 9
Page 6