UNIT 6 TTRRIIAANNGGLLEESS (A) Main Concepts and Results • The six elements of a triangle are its three angles and the three sides. • The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians. • The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes. • An exterior angle of a triangle is formed, when a side of a triangle is produced. • The measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. • The sum of the three angles of a triangle is 180°. • A triangle is said to be equilateral, if each of its sides has the same length. • In an equilateral triangle, each angle has measure 60°. • A triangle is said to be isosceles if at least two of its sides are of same length. • The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. • The difference of the lengths of any two sides of a triangle is always smaller than the length of the third side.
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Unit 6 Triangles - Tiwari Academythe measures of its two interior opposite angles. • The sum of the three angles of a triangle is 180°. • A triangle is said to be equilateral,
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UNIT 6
TTTRRRIIIAAANNNGGGLLLEEESSS
(A) Main Concepts and Results
• The six elements of a triangle are its three angles and the three
sides.
• The line segment joining a vertex of a triangle to the mid point of its
opposite side is called a median of the triangle. A triangle has
3 medians.
• The perpendicular line segment from a vertex of a triangle to its
opposite side is called an altitude of the triangle. A triangle has
3 altitudes.
• An exterior angle of a triangle is formed, when a side of a triangle is
produced.
• The measure of any exterior angle of a triangle is equal to the sum of
the measures of its two interior opposite angles.
• The sum of the three angles of a triangle is 180°.
• A triangle is said to be equilateral, if each of its sides has the same
length.
• In an equilateral triangle, each angle has measure 60°.
• A triangle is said to be isosceles if at least two of its sides are of same
length.
• The sum of the lengths of any two sides of a triangle is always greater
than the length of the third side.
• The difference of the lengths of any two sides of a triangle is always
smaller than the length of the third side.
154 EXEMPLAR PROBLEMS
MATHEMATICS
• In a right-angled triangle, the side opposite to the right angle is
called the hypotenuse and the other two sides are called its legs or
arms.
• In a right-angled triangle, the square of the hypotenuse is equal to
the sum of the squares on its legs.
• Two plane figures, say, F1 and F
2 are said to be congruent, if the
trace-copy of F1 fits exactly on that of F
2. We write this as F
1 ≅ F
2.
• Two line segments, say ABand CD , are congruent, if they have equal
lengths. We write this as ≅AB CD . However, it is common to write it
as =AB CD .
• Two angles, say ∠ABC and ∠PQR, are congruent, if their measures
are equal. We write this as ∠ABC ≅ ∠ PQR or as m ∠ABC = m∠PQR orsimply as ∠ ABC = ∠ PQR.
• Under a given correspondence, two triangles are congruent, if the
three sides of the one are equal to the three sides of the other (SSS).
• Under a given correspondence, two triangles are congruent if twosides and the angle included between them in one of the triangles
are equal to the two sides and the angle included between them ofthe other triangle (SAS).
• Under a given correspondence, two triangles are congruent if twoangles and the side included between them in one of the trianglesare equal to the two angles and the side included between them of
the other triangle (ASA).
• Under a given correspondence, two right-angled triangles are
congruent if the hypotenuse and a leg (side) of one of the trianglesare equal to the hypotenuse and one of the leg (side) of the othertriangle (RHS).
(B) Solved Examples
In Examples 1 to 5, there are four options, out of which only one is
correct. Write the correct one.
Example 1: In Fig. 6.1, side QR of a ∆PQR has been produced to the
point S. If ∠PRS = 115° and ∠P = 45°, then ∠Q is equal to,
(a) 70° (b) 105° (c) 51° (d) 80°
TRIANGLES 155
UNIT 6
Solution: Correct answer is (a).
Example 2: In an equilateral triangle ABC (Fig. 6.2), AD is an altitude.
Then 4AD2 is equal to
(a) 2BD2 (b) BC2 (c) 3AB2 (d) 2DC2
Solution: Correct answer is (c).
Example 3: Which of the following cannot be the sides of a triangle?
(a) 3 cm, 4 cm, 5 cm (b) 2 cm, 4 cm, 6 cm
(c) 2.5 cm, 3.5 cm, 4.5 cm (d) 2.3 cm, 6.4 cm, 5.2 cm
Solution: Correct answer is (b).
Fig. 6.1
Fig. 6.2
Vo
ca
bu
la
ry
1. The world equilateral contains the roots equi,which means “equal,” and lateral, whichmeans “of the side.” What do you suppose
an equilateral is?
2. The Greek prefix poly means “many,” andthe root gon means “angle.” What do yousuppose a polygon is?
156 EXEMPLAR PROBLEMS
MATHEMATICS
Example 4: Which one of the following is not a criterion for
congruence of two triangles?
(a) ASA (b) SSA (c) SAS (d) SSS
Solution: Correct answer is (b).
Example 5: In Fig. 6.3, PS is the bisector of ∠P and PQ = PR. Then
∆PRS and ∆PQS are congruent by the criterion
(a) AAA (b) SAS (c) ASA (d) both (b) and (c)
Fig. 6.3
Solution : Correct answer is (b).
In examples 6 to 9, fill in the blanks to make the statements true.
Example 6: The line segment joining a vertex of a triangle to the
mid-point of its opposite side is called its __________.
Solution: median
Example 7: A triangle is said to be ________, if each one of its sideshas the same length.
Solution: equilateral
Example 8: In Fig. 6.4, ∠ PRS = ∠ QPR + ∠ ________
Fig. 6.4
Solution: PQR
TRIANGLES 157
UNIT 6
Example 9: Let ABC and DEF be two triangles in which AB = DE,
BC = FD and CA = EF. The two triangles are congruent
under the correspondence
ABC ↔ ________
Solution: EDF
In Examples 10 to 12, state whether the statements are True or False.
Example 10: Sum of any two sides of a triangle is not less than the
third side.
Solution: False
Example 11: The measure of any exterior angle of a triangle is equal
to the sum of the measures of its two interior opposite
angles.
Solution: True
Example 12: If in ∆ABC and ∆DEF, AB = DE, ∠A = ∠D and BC = EF
then the two triangle ABC and DEF are congruent bySAS criterion.
Solution: False
Application on Problem Solving Strategy
Example 13In Fig. 6.5, find x and y.
Solution : Understand and Explore the Problem
• What all are given?
∠ABD = 60°, ∠BAD = 30° and ∠ACD = 45°
• What are to be found?
∠ADC and ∠XAC, which are respectively exterior anglesfor ∆ABD and ∆ABC.
Fig. 6.5
158 EXEMPLAR PROBLEMS
MATHEMATICS
Plan a Strategy
• Find ∠ADC using exterior angle property for ∆ABD.
• Find y using exterior angle property for ∆ABC.
Solve
• x = ∠ADC = ∠DBA + ∠BAD (In ∆ABD)
= 60° + 30°
= 90°
• y = ∠XAC = ∠ABC + ∠ACB ( In ∆ABC)
= 60° + 45°
= 105°
Revise
• Verify your answer by using some other properties of triangle.
In ∆ABD, ∠ADB = 180° – (30° + 60°) = 90° (Angle sum property
of a triangle)
x = ∠ADC = 180° – ∠ADB
= 180° – 90° = 90°, Hence, ∠ADC = 90° verified.
∠DAC = 180° – (x + 45°) = 180° – 135° = 45°
At point A on BAX������
, 30° + ∠DAC + y = 180°
Hence for verifying value of y, 30° + 45° + y = 180°
or y = 180° – 75° = 105°
Think and Discuss
1. If AD = DC? Why?
2. In given problem, can ∠B be 85° instead of 60°? If yes find the values of
x and y in that case.
3. What type of triangle is ∆ADC?
TRIANGLES 159
UNIT 6
(C) Exercise
In each of the questions 1 to 49, four options are given, out of which
only one is correct. Choose the correct one.
1. The sides of a triangle have lengths (in cm) 10, 6.5 and a, where a is
a whole number. The minimum value that a can take is
(a) 6 (b) 5 (c) 3 (d) 4
2. Triangle DEF of Fig. 6.6 is a right triangle
with ∠E = 90°.
What type of angles are ∠D and ∠F?
(a) They are equal angles
(b) They form a pair of adjacent angles
(c) They are complementary angles
(d) They are supplementary angles
3. In Fig. 6.7, PQ = PS. The
value of x is
(a) 35° (b) 45°
(c) 55° (d) 70°
4. In a right-angled triangle,
the angles other than the
right angle are
(a) obtuse (b) right
(c) acute (d) straight
5. In an isosceles triangle, one angle is 70°. The other two angles are of
(i) 55° and 55° (ii) 70° and 40° (iii) any measure
Fig. 6.6
CONGRUENT TRIANGLES
Diagram Statement Corresponding
Angles
Corresponding
Sides
∆ ABC ≅ ∆DEF ∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
≅
≅
≅
AB
BC
AC
DE
EF
DF
Fig. 6.7
160 EXEMPLAR PROBLEMS
MATHEMATICS
In the given option(s) which of the above statement(s) are true?
(a) (i) only (b) (ii) only (c) (iii) only (d) (i) and (ii)
6. In a triangle, one angle is of 90°. Then
(i) The other two angles are of 45° each
(ii) In remaining two angles, one angle is 90° and other is 45°
(iii) Remaining two angles are complementary
In the given option(s) which is true?
(a) (i) only (b) (ii) only (c) (iii) only (d) (i) and (ii)
7. Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is