Free Lines and Angles Subjective Test 02 Practice Test - 7th grade
Question 1
What do we mean by supplementary angles? [1 MARK]
SOLUTION
Solution :Two angles are supplementary if their sum is equal to 180∘.
Question 2
Find the complement of the following angles. [2 MARKS]
(a) 65∘
(b) 32∘
SOLUTION
Solution :Each Part: 1 Mark
Sum of complementary angle is equal to 90∘.
Therefore, the complement of:
(a) 65∘
⇒90∘−65∘=25∘
(b) 32∘
⇒90∘−32∘=58∘
Question 3
(a) Find the angle which is supplementary to itself.
(b) Find the angle which is double its supplement.
[2 MARKS]
SOLUTION
Solution :Solution: 1 Mark each
(a) Let the angle be x.
therefore its supplement = x (as the supplementary angles are equal)
Sum of supplementary angles is 180∘
x+x=180∘
⇒x=90∘
(b) Let the angle be x.
Its supplement = 2x.
Sum of supplementary angles is 180∘
∴x+2x=180∘
⇒3x=180∘
⇒x=60∘
∴ The angle which is double its supplement = 2x = 120∘
Question 4
(a) What kind of numbers do you get if you add consecutive triangular numbers? Give examples.
(b) Complement of ∠E is ∠F, if ∠E= 28∘, find ∠F.
[2 MARKS]
SOLUTION
Solution :Each part: 1 Mark
(a) By adding consecutive triangular numbers, we get square numbers.
e.g. 1 + 3 = 4, 6 + 10 = 16, etc.
(b) ∠F=90∘−28∘ (Sum of complementary angles is 90∘)
=62∘
Question 5
In the given figure, show that ∠1+∠3+∠8+∠6=3600. [2 MARKS]
SOLUTION
Solution :Proof: 2 Marks
∠1+∠2+∠3+∠4=3600 [Angles about a point]
∠4=∠8 [Pair of corresponding angles]
and ∠2=∠6 [Pair of corresponding angles]
⇒∠1+∠3+∠8+∠6=3600
Question 6
How are angles classified based on their measure? [3 MARKS]
SOLUTION
Solution :Acute Angle: 1 Mark
Right Angle: 1 Mark
Obtuse Angle: 1 Mark
Angles are classified based on their measure as:
(i) Acute Angle: An angle that is > 0 and < 90∘ is called an acute angle.
(ii) Right Angle: An angle that is exactly equal to 90∘, is called a right angle.
(iii) Obtuse Angle: An angle that is > 90∘ and < 180∘, is called an obtuse angle.
Question 7
(a) What are the three conditions that must be satisfied for two angles to be adjacent? [3 MARKS]
(b) In the given figure, ∠1 and ∠2 are supplementary angles. If ∠1 decreases, then ∠2 will _______ so that both the angles are still supplementary.
SOLUTION
Solution :(a) Conditions: 2 Marks
(b) Explanation: 1 Mark
(a) Two angles are said to be adjacent if they have:
i) Common vertex
ii) Common arm
iii) Non-common arms are on either side of the common arm
(b) If ∠1 decreases then, ∠2 will increase by the same measure, so that both the angles are still supplementary.
Question 8
(a) Can a triangle have two obtuse angles? Justify your answer.
(b) In a △ABC the side BC is extended to D. Given that ∠ABC=30∘ and ∠ACD=160∘. Find ∠BAC. [3 MARKS]
SOLUTION
Solution :(a) Answer: 1 Mark
Reason: 1 Mark
(b) Solution: 1 Mark
(a) A triangle can have only one obtuse angle.
If we have more than one obtuse angle in a triangle, then the sum of two (obtuse) angles will exceed 180∘.
Since the sum of all angles of a triangle is 180∘.
⇒ A triangle cannot have more than one obtuse angle.
(b)
∠BCD=180∘ (Straight angle)
∴∠ACB=∠BCD−∠ACD=(180−160)∘=20∘
⇒∠BAC=180∘−(∠ABC+∠ACB)=180−(30+20)=180−50=130
∴∠BAC=130∘
Question 9
(a) There are two sets of parallel lines passing through a plane. A transversal cuts the two sets of lines. Find the number of distinct points of intersection.
(b) The following diagram shows parallel lines cut by a transversal. Find x. [3 MARKS]
SOLUTION
Solution :(a) Solution: 1 Mark
(b) Application of concept: 1 Mark
Correct Answer: 1 Mark
(a) There are 4 points of intersection between the parallel lines & transversal.
(b) In the above figure since the two lines are parallel and cut by a transversal, using the property of corresponding angles:=> 2x - 60∘ = x
=> x - 60∘ = 0
=> x = 60∘
Question 10
(a) Find the value of x, if l is parallel to m and a is parallel to b.
(b)Find the value of x.
[4 MARKS]
SOLUTION
Solution :Each part: 2 Marks
(a) Given, a∥b and l∥m
Mark an angle 1 in the figure.
∠1=80∘ [Corresponding angles]
∠1 + ∠x= 180∘ [Since l∥m,∠1 and ∠x are cointerior angles]
⇒∠x=180∘−80∘
⇒∠x=100∘
(b) ∠5=5x+35 (Alternate angles)
45∘+∠5=180∘ (Linear pair)
∠5=180∘−45∘
=135∘
∴5x+35=135
⇒5x=135−35=100
⇒x=1005=20
Question 11
(a) Find the number of line segments in the following figure.
(b) Find the values of angles x, y and z in each of the following:
[4 MARKS]
SOLUTION
Solution :(a) Correct answer: 1 Mark
(b) Each angle: 1 Mark
(a)
The total number of line segments is 8. The line segments are AF, AB, BF, FD, FC, CD, AC and BD.
(b) (i) ∠x and ∠55∘ are vertically opposite,
∠x = 55∘
∠x + ∠z = 180∘
55∘ + ∠y = 180∘
∠y = 180∘ – 55∘ = 125∘
∠y = ∠z (Vertically Opposite angles)
∠z = 125∘
(ii) ∠z = 40∘ (Vertically opposite angles)
∠y + ∠z = 180∘
∠y = 180∘ – 40∘ = 140∘
40∘ + ∠x + 25∘ = 180∘
65∘ + ∠x = 180∘
∠x = 180∘ – 65∘ = 115∘
Question 12
From the given figure, find ∠ADG and ∠IEJ. ABCD is a rectangle. [4 MARKS]
SOLUTION
Solution :Steps: 2 Marks
Each angle: 1 Mark
Since ABCD is a rectangle, AB∥CD
∠EDC=∠EIJ=50∘ (Corresponding Angles).
Since ABCD is rectangle, ∠ADC=90∘
⇒∠ADG+∠GDC=90∘
∠ADG=90∘−50∘=40∘.
∠EJI=180∘−105∘=75∘ (Linear pair);
∠IEJ=180∘−(75∘+50∘) (Angle sum property)
⇒∠IEJ=55∘.
Question 13
In the given figure, ΔAEC and ΔDBF are equilateral. Prove that all the other triangles are also equilateral. (Given that bases of the triangles are parallel). [4 MARKS]
SOLUTION
Solution :Steps: 3 Marks
Proof: 1 Mark
ΔAEC and ΔDBF are equilateral, therefore angle made at each vertex is =60∘
Consider ΔAMN,∠MAN=60∘
∠AEC=∠AMN=60∘ (Corresponding Angles as FB∥EC)
Similarly ∠ACE=∠ANM=60∘ (Corresponding Angles).
Hence all angles of ΔAMN are equal to 60∘ and is an equilateral triangle.
Consider ΔBNO, ∠FBD=60∘∠BNO=∠ANM=60∘ (vertically Opposite Angles)
∠BON=60∘ (Angle sum property of triangle).Hence ΔBNO is also equilateral.
Similarly, we can prove it for the remaining triangles
Question 14
(a) In the given figure, the base of the triangle is parallel to line l and ∠1=∠2. Prove that line BC & m are parallel to each other.
(b) Consider the following figure. Find the angle y.
[4 MARKS]
SOLUTION
Solution :(a) Steps: 1 Mark
Proof: 1 Mark
(b) Steps: 1 Mark
Correct Answer: 1 Mark
∠1=∠2 (given)
They are also corresponding angles.
⇒l∥m...(i) (If corresponding angles are equal, then the lines are parallel)
Also, l∥BC....(ii)
From (i) and (ii):
m∥l∥BC.
⇒m∥BC.
(b) In ΔBDA, x + x + y = 180∘ (by angle sum property)
similarly, in ΔBDC, x + x + ∠BDC = 180∘
Comparing the two equations, we have ∠BDC = y.
Also, ∠BDC + ∠BDA = 180∘ (linear pair)
or, y + y = 180∘
or, y = 90∘
Question 15
In the adjoining figure, p∥q. Find the unknown angles. [4 MARKS]
SOLUTION
Solution :Application of theorems: 1 Mark
Steps: 1 Mark
Correct answers: 2 Marks
Given p ∥ q
125∘+e=180∘ [Linear pair]
∴e=180∘−125∘=55∘ (i)
e=f=55∘ [Vertically opposite angles]
a=f=55∘ [Alternate interior angles]
a+b=180∘ [Linear pair]
⇒55∘+b=180∘ [From eq. (i)]
⇒b=180∘−55∘=125∘
a=c=55∘ , b=d=125∘ [Vertically opposite angles]
⇒a=55∘, b=125∘
c=55∘, d=125∘, e=55∘, f=55∘.