VSAQS
Page-12.88Question 1:
In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.
Answer 1:
It is given that
Since, the triangles ABC and DEF are congruent, therefore,
RD Sharma 2020 solution class 9 chapter 12 Congruent Triangles VSAQS
Question 2:
In two triangles ABC and DEF, it is given that ∠A = ∠D, ∠B = ∠E and ∠C =∠F. Are the two triangles necessarily congruent?
Answer 2:
It is given that
For necessarily triangle to be congruent, sides should also be equal.
Question 3:
If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ∠C = 75°, DE = 2.5 cm, DF = 5cm and ∠D = 75°. Are two triangles congruent?
Answer 3:
It is given that
Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.
Question 4:
In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?
Answer 4:
The given information and corresponding figure is given below
From the figure, we have
And,
Hence, triangles ABC and ADC are congruent to each other.
Question 5:
In triangles ABC and CDE, if AC = CE, BC = CD, ∠A = 60°, ∠C = 30° and ∠D = 90°.
Are two triangles congruent?
Answer 5:
For the triangles ABC and ECD, we have the following information and corresponding figure:
In triangles ABC and ECD, we have
The SSA criteria for two triangles to be congruent are being followed. So both the triangles are congruent.
Question 6:
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
Answer 6:
In the triangle ABC it is given that
, 
and 
are medians.
We have to show that 
To show we will show that 
In triangle ΔBFC and ΔBEC
As, so
.........(1)
BC=BC (common sides) ........(2)
Since,
As F and E are mid points of sides AB and AC respectively, so
BF = CE ..........(3)
From equation (1), (2), and (3)
Hence
Proved.
Question 7:
Find the measure of each angle of an equilateral triangle.
Answer 7:
In equilateral triangle we know that each angle is equal
So
Now 
(by triangle property)
Hence
.
Question 8:
CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ΔADE ≅ ΔBCE.
Answer 8:
We have to prove that 
Given 
is a square
So 
Now in
is equilateral triangle.
So 
In 
and 
(Side of triangle)
(Side of equilateral triangle)
And,
So 
Hence from
congruence 
Proved.
Question 9:
Prove that the sum of three altitudes of a triangle is less than the sum of its sides.
Answer 9:
We have to prove that the sum of three altitude of the triangle is less than the sum of its sides.
In 
we have
, 
and 
We have to prove
As we know perpendicular line segment is shortest in length
Since
So 
........(1)
And
........(2)
Adding (1) and (2) we get
........(3)
Now, so
.......(4)
And again, this implies that
........(5)
Adding (3) & (4) and (5) we have
Hence
Proved.
Question 10:
In the given figure, if AB = AC and ∠B = ∠C. Prove that BQ = CP.
Answer 10:
It is given that
, and 
We have to prove that
We basically will prove 
to show
In 
and 
(Given)
(Given)
And 
is common in both the triangles
So all the properties of congruence are satisfied
So
Hence 
Proved.
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