4. In ABC, B =90° A=30° C=60 In ABC B=90, A 30° (= 60° Ву theorem AB= XAC.
Answers
• 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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• 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 F2 are said to be congruent, if the
trace-copy of F1 fits exactly on that of F2. We write this as F1 ≅ F2.
• 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 or
simply 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 two
sides and the angle included between them in one of the triangles
are equal to the two sides and the angle included between them of
the other triangle (SAS).
• Under a given correspondence, two triangles are congruent if two
angles and the side included between them in one of the triangles
are 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 triangles
are equal to the hypotenuse and one of the leg (side) of the other
triangle (RHS).
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°
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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
1. The word equilateral contains the roots equi,
which means “equal,” and lateral, which
means “of the side.” What do you suppose
an equilateral is?
2. The Greek prefix poly means “many,” and
the root gon means “angle.” What do you
suppose a polygon is?
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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 sides
has the same length.
Solution: equilateral
Example 8: In Fig. 6.4, ∠ PRS = ∠ QPR + ∠ ________
Fig. 6.4
Solution: PQR
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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 by
SAS criterion.
Solution: False
Example 13
In 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 angles
for ∆ABD and ∆ABC
Step-by-step explanation:
Answer:
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Step-by-step explanation:
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