Download Grade 7 Triangle and its properties

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Grade 7
Triangle and its properties
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Answer t he quest ions
(1)
Find missing angle
(2)
If AB and CD are parallel, f ind the value of x+y.
(3)
Find sum of f ollowing angles,
∠a + ∠b + ∠c + ∠d + ∠e + ∠f = ?
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(4) Find missing angle
(5)
If a triangle ABC is drawn so that sides AB and AC are equal and ∠C is 15°, then what is the
value of angle A?
(6) Find sum of f ollowing angles,
∠pqr + ∠qrs + ∠rst + ∠stu + ∠tup+ ∠upq = ?
Choose correct answer(s) f rom given choice
(7)
If angle ∠A is a right angle, f ind perimeter of the triangle (All
dimensions are in meters).
(8)
a. 436 meters
b. 431 meters
c. 432 meters
d. 435 meters
T he hypotenuse of a right angle triangle is 90 meters long. If one of the remaining side is 54
meters long, what is the length of another side?
a. 70 meters
b. 75 meters
c. 72 meters
d. 76 meters
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(9) If line AC and DE are parallel to each other, f ind value of angle x ?
a. 65 °
b. 55 °
c. 60 °
d. 115 °
(10)
If distance between B and D is 90 cm, f ind the distance
between A and C.
a. 71 cm
b. 72 cm
c. 80 cm
d. 74 cm
Fill in t he blanks
(11)
If angles in a triangle are p - 10, p + 20 and p + 50, value of p =
?
(12) A triangle has three angles, which we call the f irst, second and third angle here. T he second
angle is twice the f irst angle. T he third angle is 138°. T he f irst angle is
(13)
Value of missing angle =
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°.
°
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ID : ww-7-Triangle-and-its-properties [4]
Check True/False
(14) Side lengths of 29 cm, 21 cm and 20 cm make a right angle triangle.
T rue
False
(15) In triangle ΔABC and ΔPQR, If ∠A = ∠Q , AC = QP and BA = RQ ,
T riangles are congruent.
T rue
False
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ID : ww-7-Triangle-and-its-properties [5]
Answers
(1)
31°
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 + 46°
⇒ 77° = x + 46°
⇒ 77° - 46° = x
⇒ x = 77° - 46°
⇒ x = 31 °
Step 4
T heref ore the missing angle is 31 °.
(2)
105 °
Step 1
According to question AB and CD are parallel,
theref ore ∠x = 45° [Alternate interior angles]
∠y = 60° [Alternate interior angles]
Step 2
Now the value of x+y = 45° + 60° = 105°
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ID : ww-7-Triangle-and-its-properties [6]
(3)
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°
(4) 48°
Step 1
In the triangle ABC,
∠C = 88° ...[Vertically opposite angles]
Step 2
Now, in triangle ABC, ∠B + ∠88° + 44° = 180° ...[T he sum of all three angles of a
triangle is equals to 180°]
⇒ ∠B = 48°
Step 3
T he missing angle = ∠B = 48° ...[Vertically opposite angles]
Step 4
T heref ore, the value of missing angle is 48°.
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(5)
150°
Step 1
Following f igure shows the triangle ABC,
An isosceles triangle is a triangle with (at least) two equal sides. According to the question,
the side AB is equals to the side AC, the triangle ABC is an isosceles triangle.
Step 2
In an isosceles triangle the angles opposite to the equal sides, are equal. Hence ∠B equals
∠C.
Since, angle C = 15°,
theref ore, ∠B = 15°.
Step 3
T he sum of all the angles of a triangle is equals to 180°.
T heref ore, ∠A + ∠B + ∠C = 180°
or ∠A + 15 + 15 = 180
or ∠A + 30 = 180
or ∠A = 180 - 30
or ∠A = 150°
Step 4
T hus, the value of the ∠A is 150°.
(6) 720 °
Step 1
If you look at the f igure caref ully, you will notice that it is a combination of 4 triangles and
the sum of angles ∠pqr + ∠qrs + ∠rst + ∠stu + ∠tup+ ∠upq is equal to the sum of the
angles of all 4 triangles.
Step 2
T he sum of all three angles of a triangle is 180°, theref ore the sum of all the angles of 4
triangles is = 4 × 180 = 720 °,
and hence the sum of the angles ∠pqr + ∠qrs + ∠rst + ∠stu + ∠tup+ ∠upq is 720 °.
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(7) c. 432 meters
Step 1
According to question in triangle ΔABC angle ∠A is a right angle, theref ore the triangle
ΔABC is a right angle triangle.
Step 2
Lets assume length of BC is x meters.
In ΔABC
BC2 = AB2 + AC2
⇒ 1802 = 1082 + x2
By solving it, x = 144.
T he length of BC is 144 meters.
Step 3
Now the perimeter of right angle triangle ΔABC = AB + AC + BC
= 108 + 144 + 180
= 432 meters
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(8)
c. 72 meters
Step 1
Lets assume the length of another side of a right angle triangle is x meters long.
Step 2
According to question the hypotenuse of a right angle triangle is 90 meters long and one of
the remaining side is 54 meters long as shown in the f ollowing right angle triangle.
Step 3
Since ΔABC is a right angle triangle,
theref ore in ΔABC
AB2 + BC2 = AC2
⇒ x2 + 54 2 = 902
⇒ x2 + 2916 = 8100
⇒ x2 = 8100 - 2916
⇒ x2 = 5184
⇒ x2 = 722
⇒ x = 72
Step 4
Now the length of another side is 72 meters.
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(9) a. 65 °
Step 1
If you look at the question caref ully, you will notice that line AC and DE are parallel to each
other and ∠BAC = 55°
∠BDE = ∠BAC [Corresponding angles]
⇒ ∠BDE = 55°
Step 2
T he sum of all the three angles of a triangles is 180°.
In triangle BDE
∠BDE + ∠BED + ∠DBE = 180°
⇒ 55° + ∠BED + 60° = 180°
⇒ ∠BED + 115° = 180°
⇒ ∠BED = 180° - 115°
⇒ ∠BED = 65°
⇒ x = 65 °
Step 3
T heref ore the value of angle x is 65 °.
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(10) b. 72 cm
Step 1
According to question the distance between B and D is 90 cm and you have to f ind out the
distance between A and C.
Lets assume the distance between A and C is x cm.
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 D and E,
BE = AB - CD = 96 - 42 = 54 cm
In ΔBED
BD2 = DE2 + BE2
⇒ 902 = x2 + 54 2
By solving it, x = 72.
Step 3
Now the distance between A and C is 72 cm.
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(11)
40
Step 1
According to question the angles in a triangle are p - 10, p + 20 and p + 50.
Step 2
T he sum of all three angles of a triangle must be 180°.
T heref ore (p - 10) + (p + 20) + (p + 50) = 180
⇒ p - 10 + p + 20 + p + 50 = 180
⇒ 3p + 60 = 180
⇒ 3p = 180 - 60
⇒p=
120
3
⇒ p = 40
Step 3
Now the value of p is 40.
(12)
14
Step 1
Let the measure of the f irst angle be a.
Step 2
We have been told that the second angle is twice the f irst angle. T his means that the
measure of the second angle is 2a.
Step 3
We have also been told that the measure of the third angle is 138°. T hus:
First angle = a
Second angle = 2a, and
T hird angle = 138°
Step 4
We know that the sum of all interior angles of a triangle is 180°. T hus we can write:
a + 2a + 138° = 180°
⇒ 3a = 180° - 138°
⇒a=
42°
3
⇒ a = 14°
Step 5
Hence, the value of the f irst angle is 14°.
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(13)
80
Step 1
If you look at the given triangle caref ully, you will notice that the two angles of the triangle
are given. Lets assume the third angle of the triangle is x.
Step 2
T he sum of all three angles of a triangle is equal to 180°.
In given triangle 59° + 41° + x = 180°
⇒ 100 ° + x = 180°
⇒ x = 180° - 100 °
⇒ x = 80 °
Step 3
Now the missing angle = angle x [Opposite angles]
and hence the value of missing angle is 80 °.
(14) T rue
Step 1
T he square of the largest side of a right angle triangle is equal to the sum of square of the
other two sides.
Step 2
T he square of the largest side f rom side lengths 21 cm, 20 cm and 29 cm can be written as:
292 = 212 + 202
⇒ 841 = 441 + 400
⇒ 841 = 841
Step 3
Now you will notice that the square of the largest side is equal to the sum of square of the
other two sides and hence the statement "Side lengths of 21 cm, 20 cm and 29 cm
make a right angle triangle" is T rue.
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(15) T rue
Step 1
T wo triangles are congruent if their corresponding sides or angles are equal, as shown
below.
Step 2
According to question in triangle ΔABC and ΔPQR, ∠A = ∠Q , AC = QP and BA = RQ
Step 3
By SAS(side angle side) Congruence Rul, triangle ΔABC is congruent to ΔPQR and hence
the correct answer is T rue.
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