Download Grade 7 Triangle and its properties

ID : au-7-Triangle-and-its-properties [1]
Grade 7
Triangle and its properties
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Answer t he quest ions
(1)
Which is the smallest angle of this triangle?
(2)
If angle ∠BDC and ∠ACB are right angles, f ind the value of angle ∠DCB.
(3)
If AB and CD are parallel, f ind the value of x+y.
(4)
If distance between A and C is 100 meters, f ind the distance
between B and D.
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ID : au-7-Triangle-and-its-properties [2]
(5)
Find the value of z ?
(6) In triangle In ΔXYZ , Q is a point on the side YZ such that XY = XQ = QZ . If ∠YXQ = 28° , f ind the
angle ∠YXZ .
Choose correct answer(s) f rom given choice
(7) Which of f ollowing is f alse about congruence of triangles?
a. T wo triangles are congruent if the three sides of the one are equal to the three
corresponding sides of the other
b. T wo triangles are congruent if two sides and the angle included between them in one of the
triangles are equal to the corresponding sides and the angle included between them of the
other triangle
c. T wo right-angled triangles are congruent if the hypotenuse of one of the triangles is equal
to the hypotenuse of the other triangle
d. T wo triangles are congruent if two angles and the side included between them in one of the
triangles are equal to the corresponding angles and the side included between them of the
other triangle
(8)
Which of f ollowing is f alse f or a triangle?
a. T wo angles are acute angles
b. Each angle is equal to 60°
c. One angle is obtuse angle
d. T wo angles are obtuse angles
(9) Find sum of f ollowing angles,
∠A + ∠B + ∠C + ∠D + ∠E + ∠F
a. 400 °
b. 360 °
c. 390 °
d. 300 °
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ID : au-7-Triangle-and-its-properties [3]
(10) If angle ∠ADC and ∠BDC are right angle, f ind value of x + y .
a. 85°
b. 95°
c. 100°
d. 90°
(11) How many triangles are there in this f igure ?
a. 8
b. 11
c. 7
d. 10
Fill in t he blanks
(12)
A mouse is walking around the periphery of right angle triangle
shown in the picture. T he mouse will travel a distance of
meters in 8 rounds.
(All unites in meters)
(13)
Value of missing angle =
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°
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ID : au-7-Triangle-and-its-properties [4]
(14)
Value of missing angle =
°
Check True/False
(15) Lengths 104 meters, 156 meters and 26 meters can be sides of a triangle.
T rue
False
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ID : au-7-Triangle-and-its-properties [5]
Answers
(1)
∠A
Step 1
T he shortest side of a triangle is always opposite to the smallest interior angle or you can
say that the smallest angle is always opposite to the shortest side of a triangle.
Step 2
If you look at the triangle caref ully, you will notice that angle ∠A is opposite to the smallest
side of the triangle.
Step 3
Now you can say that the smallest angle of this triangle is ∠A.
(2)
60 °
Step 1
We know that the sum of all interior angles of a triangle is 180°.
Step 2
T heref ore, f or triangle ΔABC,
∠CBD + ∠BCA + ∠CAD = 180°
⇒ ∠CBD + 90° + 60° = 180°
⇒ ∠CBD = 180° - (60° + 90°)
⇒ ∠CBD = 30°
Step 3
Similarly, f or triangle ΔCDB,
∠CBD + ∠DCB + ∠CDB = 180°
⇒ 30° + ∠DCB + 90° = 180°
⇒ ∠DCB = 180° - (30° + 90°)
⇒ ∠DCB = 60°
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ID : au-7-Triangle-and-its-properties [6]
(3)
100 °
Step 1
According to question AB and CD are parallel,
theref ore ∠x = 60° [Alternate interior angles]
∠y = 40° [Alternate interior angles]
Step 2
Now the value of x+y = 60° + 40° = 100°
(4) 125 meters
Step 1
According to question the distance between A and C is 100 meters and you have to f ind out
the distance between B and D.
Lets assume the distance between B and D is x meters.
Step 2
Lets connect the point B to point D as shown in the f ollowing f igure.
If you look at the f igure caref ully, you will notice that ΔBED is a right angle triangle and the
distance between A and C is equal to the distance between B and E,
DE = CD - AB = 119 - 44 = 75 meters
In ΔBED
BD2 = BE2 + DE2
⇒ x2 = 1002 + 752
By solving it, x = 125.
Step 3
Now the distance between B and D is 125 meters.
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ID : au-7-Triangle-and-its-properties [7]
(5)
45°
Step 1
In this f igure, it is given that ∠BDC = 85°
T heref ore, x + 85° = 180° ...[Sum of linear pair is 180°]
⇒ x = 180° - 85° = 95°
Step 2
Now, z + 40° + x = 180° ...[T he sum of all three angles of a triangle is 180°]
⇒ z + 40° + 95° = 180°
⇒ z = 180° - 135°
⇒ z = 45°
Step 3
Hence, z = 45°
(6) 66°
(7) c. T wo right-angled triangles are congruent if the hypotenuse of one of the triangles is equal to
the hypotenuse of the other triangle
T wo triangles are congruent if two angles and the side included between them in one of the
triangles are equal to the corresponding angles and the side included between them of the
other triangle as shown in the given f igure and hence the statement 'T wo right-angled
triangles are congruent if the hypotenuse of one of the triangles is equal to the
hypotenuse of the other triangle' is f alse.
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ID : au-7-Triangle-and-its-properties [8]
(8)
d. T wo angles are obtuse angles
Step 1
T he sum of all three angles of a triangle must be 180°.
Step 2
Now if you look at the all of the options caref ully, you will notice that the statement "T wo
angles are obtuse angles", can't satisf y the condition of a triangle and hence the
statement "T wo angles are obtuse angles" is f alse.
(9) b. 360 °
Step 1
If you look at the given f igure caref ully, you will notice that it is a combination of triangle
ACE and triangle BDF.
Step 2
Since the sum of all three angles of a triangle is 180°.
T heref ore ∠A + ∠C + ∠E = 180°,
∠B + ∠D + ∠F = 180°
Step 3
Now the sum of the angles ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 180° + 180° = 360°
(10) d. 90°
Step 1
We know that sum of all internal angles of a triangle is 180°, theref ore f or triangle ΔADC,
∠CAD + ∠ADC + ∠ACD = 180°
45° + 90° + x = 180°
Step 2
T heref ore x = 180° - 90° - 45°,
x = 45°
Step 3
Similarly in ΔCDB, y can be calculated as y = 180° - 90° - 45°,
y = 45°
Step 4
T heref ore, x + y = 45° + 45°
x + y = 90°
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ID : au-7-Triangle-and-its-properties [9]
(11) d. 10
Step 1
Following triangles are there in this f igure
Step 2
T heref ore there are 10 triangles in this f igure
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ID : au-7-Triangle-and-its-properties [10]
(12)
1008
Step 1
According to question a mouse is walking around the periphery of right angle triangle
shown in the picture.
Step 2
Lets assume length of BC is x meters.
In ΔABC
AC2 = AB2 + BC2
⇒ 532 = x2 + 282
By solving it, x = 45.
Step 3
Now the length of BC is 45 meters.
T he periphery of right angle triangle ΔABC = AB + BC + AC
= 28 + 45 + 53
= 126 meters
Step 4
Distance traveled by mouse in 8 rounds = 8 × 126 = 1008 meters
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ID : au-7-Triangle-and-its-properties [11]
(13)
53
Step 1
Lets assume the missing angle is x.
Step 2
T he exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
Step 3
If you look at the given triangle caref ully, you will notice that the exterior angle of the given
triangle = x + 33°
⇒ 86° = x + 33°
⇒ 86° - 33° = x
⇒ x = 86° - 33°
⇒ x = 53 °
Step 4
T heref ore the missing angle is 53 °.
(14)
80
Step 1
If you look at the triangle caref ully, you will notice the missing angle is the exterior angle of
the triangle.
Step 2
T he exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
Step 3
Now the missing angle = 47° + 33° = 80 °
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ID : au-7-Triangle-and-its-properties [12]
(15) False
Step 1
T he sum of the lengths of two sides of a triangle must always be greater than the length
of the third side.
or you can say that if lengths 104 meters, 26 meters and 156 meters satisf y the f ollowing
conditions
26 + 156 > 104,
104 + 156 > 26,
104 + 26 > 156,
T hen the lengths 104 meters, 26 meters and 156 meters can be sides of a triangle else
lengths 104 meters, 26 meters and 156 meters can not be sides of a triangle.
Step 2
All of the three conditions are not satisf ied by the lengths 104 meters, 26 meters and 156
meters and hence the statement "lengths of 104 meters, 26 meters and 156 meters
can be sides of a triangle" is False.
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