Download Congruent and similar triangles

Congruent and similar triangles
Important notation in geometric diagrams:
• Sides with the same markings are equal in length.
• Sides with the same number of arrows are parallel.
• The symbol || means ‘is parallel to’.
• The symbol ⊥ means ‘is perpendicular to’.
• The symbol ≡ means is ‘congruent to’.
• The symbol ∴ means ‘therefore’.
• The symbol ||| means ‘is similar to’.
• The triangle with vertices A, B and C is written as ΔABC.
• Angles are denoted using the symbol ∠ ; the angle at vertex A is written as
∠ BAC. They can also be given letter names such as α, β, ϕ.
• Intervals are named according to their end points; the interval between
A and B is written as AB.
Congruency
Congruent figures are identical in size and shape,
but their orientation may be different.
Tests for congruent triangles
Triangles are congruent if they are the same shape and
size. Triangles are congruent if any one of these four sets
of conditions is met:
1 Side Side Side (SSS)
The three pairs of sides have the same length.
2 Side Angle Side (SAS)
Two pairs of sides have the same length and the
included angle is the same.
3 Angle Angle Side (AAS)
Two pairs of angles are the same and one pair
of matching sides is the same length.
4 Right angle Hypotenuse Side (RHS)
If both triangles contain a right angle, the
hypotenuse in both is the same and another
pair of sides is the same length.
Worked Example 1
Show that ΔABC and ΔDFE are
congruent, stating the congruency
test used.
Thinking
1 List the sides and angles.
2 Justify your answer.
Working
AB = DF
(both 14 cm)
∠ABC = ∠DFE
(both 65°)
BC = EF
(both 9 cm)
The triangles are congruent due to SAS.
Similarity
Similar shapes are those where all side lengths have
undergone the same dilation. Similar figures have the
same shape but are different sizes. All matching sides
of one figure have been enlarged or reduced by the
same factor to form the second figure. This factor is
called the ‘dilation factor’.
When a figure is dilated:
• matching angles are the same
• matching side lengths are in the same ratio.
Tests for similar triangles
Triangles are similar if one is a proportional dilation of the other. They have the same shape but are different
sizes. Two triangles are similar if they satisfy any of the following conditions:
2 Side Angle Side (SAS)
1 Side Side Side (SSS)
If the three sides of one triangle are proportional to
the three sides of another triangle.
3 Angle Angle Angle (AAA)
If two angles of one triangle are equal to two angles
in another triangle. All three angles are therefore
equal.
If two sides of one triangle are proportional to two
sides of another triangle, and the included angles are
equal.
4 Right Angle Hypotenuse Side (RHS)
If the hypotenuse and one side of a right-angled
triangle are proportional to the hypotenuse and one
side of another right-angled triangle.
Note: These tests are similar to the four tests for congruence, the difference being that
where congruence requires that matching sides be the same length, similarity requires that matching sides
be proportional.
Finding lengths using similarity
If two triangles are similar we can form an equation to solve for the unknown length using the ratio of matching
sides. We need to identify the matching sides to equate the ratios.
Worked Example 2
Given that these pairs of triangles are similar, find x.
(a)
(b)
Thinking
Working
(a) 1 Identify matching sides and equate the two
ratios.
2 Write the ratios as an equation with x as a
numerator and solve for x.
(b) 1 Identify matching sides and equate the two
ratios.
2 Write the ratios as an equation with x as a
numerator and solve for x.
(a) x : 8 = 45 : 15
x8
= 4515
x
= 4515
x
= 24 units
×8
(b) x : 25 = 6 : 18
x25
= 618
x
= 618
× 25
x
= 813
units
3 Write the length as an exact measurement. Do
not approximate with decimals unless instructed to do
so.
To find the length of an unknown side in a similar triangle:
1 Find two pairs of matching sides that include the unknown side.
2 Equate the two ratios.
3 Solve the equation to find the exact value of the unknown, unless told otherwise.
Worked Example 1
Show that ΔABC and ΔDFE are
congruent, stating the congruency
test used.
Thinking
1 List the sides and angles.
2 Justify your answer.
Working
AB = DF
(both 14 cm)
(both 65°)
∠ ABC = ∠ DFE
BC = EF
(both 9 cm)
The triangles are congruent due to SAS.
1 Show that the following pairs of triangles are congruent, stating the congruency test used.
(a)
(c)
(e)
(b)
(d)
(f)
(g)
(h)
Worked Example 2
Given that these pairs of triangles are similar, find x.
(a)
(b)
Thinking
(a) 1 Identify matching sides and equate the two
ratios.
2 Write the ratios as an equation with x as a
numerator and solve for x.
(b) 1 Identify matching sides and equate the two
ratios.
2 Write the ratios as an equation with x as a
numerator and solve for x.
Working
(a) x : 8 = 45 : 15
x8
=
4515
x
=
4515 × 8
x
= 24 units
(b) x : 25 = 6 : 18
x25
=
618
x
=
618 × 25
x
= 813
3 Write the length as an exact measurement. Do
not approximate with decimals unless instructed to do
so.
units
2 Given that these pairs of triangles are similar, find x.
(a)
(b)
(c)
(d)
(e)
(f)
3 Match the pairs of congruent triangles given below.
(a)
(b)
(d)
(e)
(c)
(f)
4 Which one of the following conditions does not ensure congruence for two triangles?
A SSS B SAS C AAS D AAA
5 Which condition verifies that these two triangles are
congruent?
A SSS
B SAS
C AAS
D RHS
6 Given ΔABC ||| ΔEFG:
(a) the matching side to AB is:
A FG
B EF
C BC
D AC
(b) the matching side to FG is:
A BC
B AC
C EF
D EG
7 Given ΔHJK ||| ΔLMN:
(a) the matching side to LM is:
A HJ
B MN
C JK
D LN
(b) the matching side to HJ is:
A JK
B LM
C HK
D NL
8 In the following diagrams, first determine the similar triangles and then find the value of x.
(a)
(c)
(b)
(d)
9 Dirk is climbing up a long ladder leaning against a wall. His
friend Sean is holding up the ladder to prevent it wobbling. Sean
is 3 m from the wall and 2 m from the base of the ladder. With
his arms stretched above his head he holds the ladder at a point
2.1 m above the ground. How far up the wall does the ladder
reach?
10 A light source projects the image of a photographic slide onto
a screen, as shown in the diagram. The slide is 3 cm high and
positioned 10 cm behind a lens. Calculate the distance between
the lens and the screen if the image on the screen is 95 cm high.
Give your answer correct to two decimal places.