SIMILARITY AND CONGRUENCY
1. Similarity
If two triangles have the same shape but have different sizes, these two triangles are said to be similar. The symbol for similar is ~
Example:
These two triangles are similar
For triangles to be similar, they are some special properties they need to have. These are:
Ø The ratio of their corresponding sides should be equal. This means that the ratio of a side in the first triangle and its corresponding side in the other triangle should be equal to the ratio of other corresponding sides. By corresponding sides we mean sides opposite to the equal angles.
Y
B
A C X Z
So, AB:XY , BC: YZ , AC:XZ
AB = BC = AC = k (constant)
XY YZ XZ
ABC ~ XYZ
Ø The angles of the two triangles should be equal.
Q
D
R S E F
Q = D R = E S = F
QRS ~ DEF
Some real life examples of similarity in objects are:
But there are also some objects that are similar to another object all the time in any case. An example of this is a circle. It is always similar to another circle because it is always 360˚.
360˚ 360˚
If you have 2 polygons other than triangles eg rectangles, trapezium, right angled triangles, they need to have both the properties to prove them similar while in a triangle, one property is enough to prove them similar.
If there are 2 rectangles and you need to prove them similar. You only have to look at the first property as the second one is proved from the start. This is because angles in a rectangle are always 90˚. So you just have to see if the ratio of each of the corresponding sides is equal.
AB = BC = CD = AC = k
WX XY YZ WZ
10 = 6 = 10 = 6 = k
5 3 5 3
k = 2 ABCD ~ WXYZ
III. THEOREMS
Similar triangles can be recognized through some theorems. These theorems are:
AA theorem (angle-angle theorem)
If two triangles have two corresponding and equal angles in each, then the two triangles are considered to be similar. This is because if these two angles are corresponding and equal, then the third will also be equal and so the triangle will have all equal angles
Example:
<A = <D <B= <E
<C = 180˚ - <B - <A <F= 180˚ - <E - <D
<C=180˚-55˚-70˚ <F= 180˚-55˚- 70˚
<C = 55˚ <F= 55˚
<C=<F
ABC ~ DEF
SAS theorem (side-angle-side theorem)
If in two triangles, the ratio of two corresponding sides is equal and the angle is also equal, then the two triangles are similar.
K = 2 <A = <X
ABC ~ XYZ
SSS theorem (side-side-side theorem)
In two triangles, if the ratio of the three corresponding sides is equal, then the two triangles are similar
Y
B
A C X Z
So, AB:XY , BC: YZ , AC:XZ
AB = BC = AC = k (constant)
XY YZ XZ
ABC ~ XYZ
Intercept theorem
If in two triangles, one side is parallel to its corresponding side in the other triangle then the ratio of the two other corresponding sides will be equal thus the two triangles would be similar.
If DE is parallel to BC (DE//BC), then AD = AE = k
AB AC
ABC ~ ADE
Converse of Intercept of theorem
This is the opposite of the previous theorem. In this, If the ratio of two corresponding sides are equal, then the 3rd side of the triangle is parallel to its corresponding side and thus the two triangles are similar.
IV. AREAS AND PERIMETERS
Perimeters
If in two triangles, the ratio of the corresponding sides is equal, then the ratio of the perimeters is also equal. This means that in two similar triangles, the ratio of the corresponding sides is equal to the ratio of the perimeters of the two triangles.
Example:
AB = AC = BC = k P= k
A1B1 A1C1 B1C1 P1
k=3 k= 9 + 15 +6
3 + 5 + 2
k= 3
AB = AC = BC = P = k
A1B1 A1C1 B1C1 P1
Area
If in two triangles, the ratio of the corresponding sides is equal, then to get the ratio of the area, you square it. This is how it has been proved
Area of triangle ABC = ½ (AB) (BC) Sin B
Area of triangle A1B1C1= ½(A1B1) (B1C1) Sin B1
A= ½ (AB) (BC) Sin B
A1=½(A1B1) (B1C1) Sin B1
Since Sin B =Sin B1
A = (AB) ×(BC)
A1= (A1B1) ×(B1C1)
Since AB = BC = k
A1B1 B1C1
A = k × k A = k2
A1 A1
V) EXERCISE
1. In the figure, given that △ABC ~ △PQR, find the unknowns x, y and z.
2. In the figure, △ABC ~ △RPQ. Find the values of the unknowns.
3. Show that △ABC and △PQR in the figure are similar.
4) Seen from earth, the sun and the moon appear to be the same size. The approximate distance from the earth and the sun and moon are 150,000,000km and 400,000km respectively. Diameter of moon is 3500 km. find diameter of sun.
4. Congruency
Triangles which are exactly the same shape and size are said to be congruent.
For two triangles to be congruent there is a condition. This condition is that all the six elements of the first triangle which are the three angles and three sides should be equal to the six elements of the second triangle.
III. THEROEMS
Congruent triangles can be recognized through some theorems. These are:
SSS theorem (side-side-side theorem)
In this theorem, if three sides from one triangle are equal to three sides from the other triangle, then the two triangles are congruent
A J
B C K L
ABC = JKL
SAS theorem (side-angle-side theorem)
If two sides and the angle between them in one triangle is equal to two sides and the angle between them in the second triangle, then the two triangles are congruent.
AAS theorem (angle-angle-side theorem)
If one side and two angles in one triangle are equal to one side and two similarly located angles in the second triangle, then the two triangles are congruent.
RHS theorem (Right angle- hypotenuse-side theorem)
If in two right angled triangles, the hypotenuse are equal, and any other side is equal, then the two triangles are said to be congruent.
IV) EXERCISE
1.Determine which of the following pair(s) of triangles are congruent
(i)
(ii)
2. Given that △ABC @ △XYZ in the figure, find the unknowns p, q and r.
5. 3-D shapes
These two objects are similar as their ration of their corresponding sides is equal. The ratio is 2.
I)SURFACE AREAS
Surface area of A= 4 cm × 3cm
Surface area of A1= 8 cm ×6cm
A= 4 × 3
A1=8 × 6
A = k × k = k2
A1
When there are two 3-D objects with corresponding sides, the ratio of the surface area of the two objects is k2 while the ratio of the corresponding sides of the two objects is k.
II) VOLUME
Volume of object A= 4cm ×3cm× 2cm
Volume of object A1= 8cm ×6cm × 4cm
A= 4 × 3 × 2
A1=8 × 6 × 4
A = k × k × k = k3
A1
When there are two 3-D objects with corresponding sides, the ratio of the volume of the two objects is k3 while the ratio of the corresponding sides of the two objects is k.
III) Exercise
1)if k=3 and the sides of a small cube are 6cm, 5cm, 4cm, find the corresponding sides of the larger cube.
2)if k=2 and the sides of a small cube are 4cm, 6cm, 8cm, find the volume of the larger cube.
Answers
Exercise on similarity
1) x = 30°y = 98°z= 7.5
2) x = 90°y = 36 z = 65
3) In △ABC and △PQR as shown,
∠B = ∠Q, ∠C = ∠R,
∠A = 180° – 35° – 75°= 70°∠P=70°
△ABC ~ △PQR (AAA)
4)131250 km
Exercise on congruency
1) (i)I and III, II and IV (ii) I and II, III and IV
2) P = 6cm, q=5cm,r=50˚
Exercise on 3D shapes
1)18cm, 15cm, 12cm
2)1536cm3
Jaffer, you have not given much examples on polygons and 3D shapes.
Your exercise problems are also very few.
You should develop using equations for writing equations.
GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCYWriting skill / Drawing skill / Formatting skill / Polygon / Triangle / 3D shapes / Questions / Examples / Total
3 / 2 / 3 / 2 / 2 / 3 / 3 / 2 / 20
creativity flow neatness / pictures labeling / page paragraphs math equations / always-similar polygons appln / theorems applns / perimeter vol, SA / 3D shapes congruency variety / ex+ans
2 / 2 / 3 / 1 / 1 / 3 / 1 / 1 / 14
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