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Scalene
Triangle
• Scalene
triangles are
triangles in which
none of the sides
are the same
length.
Isosceles
Triangle
• Isosceles
triangles are
triangles in which
two of the sides
are the same
length.
Equilateral
Triangle
• Equilateral
triangles are
triangles in which
all three sides are
the same length.
Types of
triangles on
the basis of
length of
sides
Acute
Triangle
• Acute
triangles are
triangles in which
the measures of
all three angles
are less than 90
degrees.
Obtuse
Triangle
• Obtuse
triangles are
triangles in which
the measure of
one angle is
greater than 90
degrees.
Right
Triangle
• Right
triangles are
triangles in which
the measure of
one angle equals
90 degrees.
Types of
triangles on
the basis of
measure of
angles
centroid.
orthocenter.
incenter.
circumcenter.
• Area = ½ × base × height
• Area = √s (s-a)(s-b) (s-c)
• Area of equilateral triangle =
{a,b and c are lengths of sides}
1. SSS (Side-Side-Side): If three pairs of
sides of two triangles are equal in length,
then the triangles are congruent.
2. SAS (Side-Angle-Side): If two pairs of sides
of two triangles are equal in length, and the
included angles are equal in measurement, then
the triangles are congruent.
3. ASA (Angle-Side-Angle): If two pairs of angles
of two triangles are equal in measurement, and
the included sides are equal in length, then the
triangles are congruent.
4. RHS (Right-angle-Hypotenuse-Side): If two
right-angled triangles have their hypotenuses
equal in length, and pair of shorter sides is equal
in length, then the triangles are congruent.
The figures which have the same shape but not
necessarily of the same size, are called similar figures.
Figures are said to be similar if :
• They are enlargements of each other
• Their corresponding angles are equal
• Their corresponding sides are proportional
Example :
(i) Every circle is similar to another circle.
(ii) Every square similar to another square
(iii) A right angle triangle is similar to any other right
angle triangle
Two triangles are similar, if
(1) their corresponding angles are equal
(2) their corresponding sides are in the same ratio (or
proportion).
If corresponding angles of two
triangles are equal, then they are known as equiangular
triangles.
c. 624 BC
c. 546 BC
Pre-Socratic
philosophy
Western Philosophy
Ionian/Milesian
school, Naturalism
AAA Similarity Criterion:
If in two triangles, the
corresponding angles are equal,
then their corresponding sides
are proportional and hence the
triangles are similar.
Corollary:
If two angles of a triangle are
respectively equal to two angles
of another triangle, then the
two triangles are similar by AA
Similarity Criterion.
SSS Similarity Criterion:
If the corresponding sides of
two triangles are
proportional their
corresponding angles are
equal and hence the
triangles are similar.
In the given triangles we
observe that the ratio of the
corresponding sides is the
same.
3/6 = 4/8 = 5/10 = 1/2
SAS Similarity Criterion:
If one angle of a triangle is
equal to one angle of the
other and the sides including
these angles are
proportional, the triangles
are similar.
Property:
If a perpendicular is drawn
from the vertex of the right
angle of a right triangle to
the hypotenuse, the
triangles on each side of the
perpendicular are similar to
the whole triangle and to
each other.
1. Which of the following triangles are always similar?
(a) right triangles
(b) isosceles triangles
(c) equilateral triangles
2. The sides of a triangle are 5, 6 and 10. Find the length of the
longest side of a similar triangle whose shortest side is 15.
(a) 10
(b) 15
(c) 18
(d) 30
3. Similar triangles are exactly the same shape and size.
(a) True
(b) False
4. Given: In the diagram, is parallel to
, BD = 4, DA = 6 and EC = 8. Find BC to
the nearest tenth.
(a) 4.3
(b) 5.3
(c) 8.3
(d) 13.3
5. Find BC.
(a) 4
(b) 4.5
(c) 13.5
(d) 17
6. Two triangles are similar. The sides of the first triangle are 7, 9,
and 11. The smallest side of the second triangle is 21. Find the
perimeter of the second triangle.
(a) 27
(b) 33
(c) 63
(d) 81
7. Two ladders are leaned against a wall
such that they make the same angle with
the ground. The 10' ladder reaches 8' up
the wall. How much further up the wall
does the 18' ladder reach?
(a) 4.5'
(b) 6.4'
(c) 14.4'
(d) 22.4‘
8. At a certain time of the day,
the shadow of a 5' boy is 8'
long. The shadow of a tree at
this same time is 28' long. How
tall is the tree?
(a) 8.5'
(b) 16'
(c) 17.5'
(d) 20‘
9. Two triangular roofs are similar. The
corresponding sides of these roofs are 16
feet and 24 feet. If the altitude of the
smaller roof is 6 feet, find the
corresponding altitude of the larger roof.
(a) 6
(b) 9
(c) 36
(d) 81
10. Two polygons are similar. If the ratio of the perimeters is 7:4,
find the ratio of the corresponding sides.
(a) 7 : 4
(b) 49 : 16
11. The ratio of the perimeters of two similar triangles is 3:7. Find
the ratio of the areas.
(a) 3 : 7
(b) 9 : 49
12. The areas of two similar polygons are in the ratio 25:81. Find the
ratio of the corresponding sides.
(a) 5 : 9
(b) 25 : 81
(c) 625 : 6561
13. Given: These two regular polygons
are similar. What is their ratio of
similitude (larger to smaller)?
(a) 8 : 12
(b) 2 : 1
(c) 4 : 8
14. Given: These two rhombi are
similar. If their ratio of similitude is 3 :
1, find x.
(a) 4
(b) 6
(c) 9
16. Given angle A and angle A' are each
59º, find AC.
(a) 8
(b) 10
(c) 12
(d) 18
17. The side of one cube measures 8
inches. The side of a smaller cube measures
6 inches. What is the ratio of the volumes of
the two cubes (larger to smaller)?
(a) 8:6
(b) 16:9
(c) 64:27
(d) 64:36
18. In triangle ABC, angle A = 90º and angle B = 35º. In triangle
DEF, angle E = 35º and angle F = 55º. Are the triangles similar?
(a) YES
(b) NO
Δ ABC ~ Δ PQR
Draw AD ⊥ BC and PM ⊥ PR
Hence;
Now, in Δ ABD and Δ PQM;
∠ A = ∠ P, ∠ B = ∠ Q and ∠ D = ∠ M (because Δ ABC ~ Δ PQR)
Hence; Δ ABD ~ Δ PQM
Hence;
Since, Δ ABC ~ Δ PQR
So,
Hence;
If a perpendicular is drawn from the vertex of the right angle of a
right triangle to the hypotenuse then triangles on both sides of the
perpendicular are similar to the whole triangle and to each other.
Construction: Triangle ABC is drawn which is
right-angled at B. From vertex B, perpendicular
BD is drawn on hypotenuse AC.
To prove:
Δ ABC ~ Δ ADB ~ Δ BDC
Proof:
In Δ ABC and Δ ADB;
∠ ABC = ∠ ADB
∠ BAC = ∠ DAB
∠ ACB = ∠ DBA
From AAA criterion; Δ ABC ~ Δ ADB
In Δ ABC and Δ BDC;
∠ ABC = ∠ BDC
∠ BAC = ∠ DBC
∠ ACB = ∠ DBC
From AAA criterion; Δ ABC ~ Δ BDC
Hence; Δ ABC ~ Δ ADB ~ Δ BDC proved.
c. 570 BC
Samos
c. 495 BC (aged
around 75)
Metapontum
Ancient Philosophy
Western philosophy
Pythagoreanism
Construction: Triangle ABC is drawn which is right
angled at B. From vertex B, perpendicular BD is drawn
on hypotenuse AC.
To prove:
Proof:
In Δ ABC and Δ ADB;
In a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides.
Because these are similar triangles
….(1)
In Δ ABC and Δ BDC;
Adding equations (1) and (2), we get;
Proved.
….(2)
1. Two figures having the same shape but not necessarily the same
size are called similar figures.
2. All the congruent figures are similar but the converse is not true.
3. Two polygons of the same number of sides are similar, if (i) their
corresponding angles are equal and (ii) their corresponding sides are
in the same ratio (i.e., proportion).
4. If a line is drawn parallel to one side of a triangle to intersect the
other two sides in distinct points, then the other two sides are
divided in the same ratio.
5. If a line divides any two sides of a triangle in the same ratio, then
the line is parallel to the third side.
6. If in two triangles, corresponding angles are equal, then their
corresponding sides are in the same ratio and hence the two
triangles are similar (AAA similarity criterion).
7. If in two triangles, two angles of one triangle are respectively
equal to the two angles of the other triangle, then the two triangles
are similar (AA similarity criterion).
8. If in two triangles, corresponding sides are in the same ratio, then
their corresponding angles are equal and hence the triangles are
similar (SSS similarity criterion).
9. If one angle of a triangle is equal to one angle of another triangle
and the sides including these angles are in the same ratio
(proportional), then the triangles are similar (SAS similarity
criterion).
10. The ratio of the areas of two similar triangles is equal to the
square of the ratio of their corresponding sides.
11. If a perpendicular is drawn from the vertex of the right angle of
a right triangle to the hypotenuse, then the triangles on both sides
of the perpendicular are similar to the whole triangle and also to
each other.
12. In a right triangle, the square of the hypotenuse is equal to the
sum of the squares of the other two sides (Pythagoras Theorem).
13. If in a triangle, square of one side is equal to the sum of the
squares of the other two sides, then the angle opposite the first
side is a right angle.
Triangles X CLASS CBSE NCERT

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Triangles X CLASS CBSE NCERT

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  • 5. Scalene Triangle • Scalene triangles are triangles in which none of the sides are the same length. Isosceles Triangle • Isosceles triangles are triangles in which two of the sides are the same length. Equilateral Triangle • Equilateral triangles are triangles in which all three sides are the same length. Types of triangles on the basis of length of sides
  • 6. Acute Triangle • Acute triangles are triangles in which the measures of all three angles are less than 90 degrees. Obtuse Triangle • Obtuse triangles are triangles in which the measure of one angle is greater than 90 degrees. Right Triangle • Right triangles are triangles in which the measure of one angle equals 90 degrees. Types of triangles on the basis of measure of angles
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  • 10. • Area = ½ × base × height • Area = √s (s-a)(s-b) (s-c) • Area of equilateral triangle = {a,b and c are lengths of sides}
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  • 12. 1. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
  • 13. 2. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
  • 14. 3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
  • 15. 4. RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have their hypotenuses equal in length, and pair of shorter sides is equal in length, then the triangles are congruent.
  • 16. The figures which have the same shape but not necessarily of the same size, are called similar figures. Figures are said to be similar if : • They are enlargements of each other • Their corresponding angles are equal • Their corresponding sides are proportional Example : (i) Every circle is similar to another circle.
  • 17. (ii) Every square similar to another square (iii) A right angle triangle is similar to any other right angle triangle
  • 18. Two triangles are similar, if (1) their corresponding angles are equal (2) their corresponding sides are in the same ratio (or proportion).
  • 19. If corresponding angles of two triangles are equal, then they are known as equiangular triangles.
  • 20. c. 624 BC c. 546 BC Pre-Socratic philosophy Western Philosophy Ionian/Milesian school, Naturalism
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  • 23. AAA Similarity Criterion: If in two triangles, the corresponding angles are equal, then their corresponding sides are proportional and hence the triangles are similar. Corollary: If two angles of a triangle are respectively equal to two angles of another triangle, then the two triangles are similar by AA Similarity Criterion.
  • 24. SSS Similarity Criterion: If the corresponding sides of two triangles are proportional their corresponding angles are equal and hence the triangles are similar. In the given triangles we observe that the ratio of the corresponding sides is the same. 3/6 = 4/8 = 5/10 = 1/2
  • 25. SAS Similarity Criterion: If one angle of a triangle is equal to one angle of the other and the sides including these angles are proportional, the triangles are similar. Property: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
  • 26. 1. Which of the following triangles are always similar? (a) right triangles (b) isosceles triangles (c) equilateral triangles 2. The sides of a triangle are 5, 6 and 10. Find the length of the longest side of a similar triangle whose shortest side is 15. (a) 10 (b) 15 (c) 18 (d) 30 3. Similar triangles are exactly the same shape and size. (a) True (b) False
  • 27. 4. Given: In the diagram, is parallel to , BD = 4, DA = 6 and EC = 8. Find BC to the nearest tenth. (a) 4.3 (b) 5.3 (c) 8.3 (d) 13.3 5. Find BC. (a) 4 (b) 4.5 (c) 13.5 (d) 17 6. Two triangles are similar. The sides of the first triangle are 7, 9, and 11. The smallest side of the second triangle is 21. Find the perimeter of the second triangle. (a) 27 (b) 33 (c) 63 (d) 81
  • 28. 7. Two ladders are leaned against a wall such that they make the same angle with the ground. The 10' ladder reaches 8' up the wall. How much further up the wall does the 18' ladder reach? (a) 4.5' (b) 6.4' (c) 14.4' (d) 22.4‘ 8. At a certain time of the day, the shadow of a 5' boy is 8' long. The shadow of a tree at this same time is 28' long. How tall is the tree? (a) 8.5' (b) 16' (c) 17.5' (d) 20‘
  • 29. 9. Two triangular roofs are similar. The corresponding sides of these roofs are 16 feet and 24 feet. If the altitude of the smaller roof is 6 feet, find the corresponding altitude of the larger roof. (a) 6 (b) 9 (c) 36 (d) 81 10. Two polygons are similar. If the ratio of the perimeters is 7:4, find the ratio of the corresponding sides. (a) 7 : 4 (b) 49 : 16 11. The ratio of the perimeters of two similar triangles is 3:7. Find the ratio of the areas. (a) 3 : 7 (b) 9 : 49
  • 30. 12. The areas of two similar polygons are in the ratio 25:81. Find the ratio of the corresponding sides. (a) 5 : 9 (b) 25 : 81 (c) 625 : 6561 13. Given: These two regular polygons are similar. What is their ratio of similitude (larger to smaller)? (a) 8 : 12 (b) 2 : 1 (c) 4 : 8 14. Given: These two rhombi are similar. If their ratio of similitude is 3 : 1, find x. (a) 4 (b) 6 (c) 9
  • 31. 16. Given angle A and angle A' are each 59º, find AC. (a) 8 (b) 10 (c) 12 (d) 18 17. The side of one cube measures 8 inches. The side of a smaller cube measures 6 inches. What is the ratio of the volumes of the two cubes (larger to smaller)? (a) 8:6 (b) 16:9 (c) 64:27 (d) 64:36 18. In triangle ABC, angle A = 90º and angle B = 35º. In triangle DEF, angle E = 35º and angle F = 55º. Are the triangles similar? (a) YES (b) NO
  • 32. Δ ABC ~ Δ PQR Draw AD ⊥ BC and PM ⊥ PR
  • 33. Hence; Now, in Δ ABD and Δ PQM; ∠ A = ∠ P, ∠ B = ∠ Q and ∠ D = ∠ M (because Δ ABC ~ Δ PQR) Hence; Δ ABD ~ Δ PQM Hence; Since, Δ ABC ~ Δ PQR So, Hence;
  • 34. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other. Construction: Triangle ABC is drawn which is right-angled at B. From vertex B, perpendicular BD is drawn on hypotenuse AC. To prove: Δ ABC ~ Δ ADB ~ Δ BDC Proof: In Δ ABC and Δ ADB; ∠ ABC = ∠ ADB ∠ BAC = ∠ DAB ∠ ACB = ∠ DBA From AAA criterion; Δ ABC ~ Δ ADB In Δ ABC and Δ BDC;
  • 35. ∠ ABC = ∠ BDC ∠ BAC = ∠ DBC ∠ ACB = ∠ DBC From AAA criterion; Δ ABC ~ Δ BDC Hence; Δ ABC ~ Δ ADB ~ Δ BDC proved.
  • 36. c. 570 BC Samos c. 495 BC (aged around 75) Metapontum Ancient Philosophy Western philosophy Pythagoreanism
  • 37. Construction: Triangle ABC is drawn which is right angled at B. From vertex B, perpendicular BD is drawn on hypotenuse AC. To prove: Proof: In Δ ABC and Δ ADB; In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Because these are similar triangles ….(1)
  • 38. In Δ ABC and Δ BDC; Adding equations (1) and (2), we get; Proved. ….(2)
  • 39. 1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). 4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. 5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).
  • 40. 7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). 8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). 9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). 10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.
  • 41. 12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.