Exercise 4.3 - Chapter 4 Geometry Term 2 6th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Miscellaneous Practice Problems
Question $1 .$
What are the angles of an isosceles right-angled triangle?
Solution:
Since it is a right-angled triangle
One of the angles is $90^{\circ}$
Other two angles are equal because it is an isosceles triangle.
Other two angles must be $45^{\circ}$ and $45^{\circ}$
Angles are $90^{\circ}, 45^{\circ}, 45^{\circ}$.

Question $2 .$
Which of the following correctly describes the given triangle.
(a) It is a right isosceles triangle.
(b) It is an acute isosceles triangle.
(c) It is an obtuse isosceles triangle.
(d) it is an obtuse scalene triangle.
Solution:
(c) It is an obtuse isosceles triangle.
 

Question $3 .$
Which of the following is not possible?
(a) An obtuse isosceles triangle

(b) An acute isosceles triangle
(c) An obtuse equilateral triangle
(d) An acute equilateral triangle
Solution:
(c) an obtuse equilateral triangle
 

Question $4 .$
If one angle of an isosceles triangle is $124^{\circ}$, then find the other angles.
Solution:
In an isosceles triangle, any two sides are equal. Also, two angles are equal.
Sum of three angles of a triangle $=180^{\circ}$
Given one angle $=124^{\circ}$
Sum of other two angles $=180^{\circ}-124^{\circ}=56^{\circ}$
Other angles are $=\frac{56}{2}=28^{\circ}$
$28^{\circ}$ and $28^{\circ}$.

Question 5 .
The diagram shows a square $\mathrm{ABCD}$. If the line segment joints $\mathrm{A}$ and $\mathrm{C}$, then mention the type of triangles so formed.

Solution:
For a square all sides are equal and each angle is $90^{\circ}$.
$\triangle \mathrm{ABC}$ and $\triangle \mathrm{ADC}$ are isosceles right-angled triangles.
 

Question 6 .
Draw a line segment $\mathrm{AB}$ of length $6 \mathrm{~cm}$. At each end of this line segment $\mathrm{AB}$, draw a line perpendicular to the line $\mathrm{AB}$. Are these lines parallel?
Solution:
Here $\mathrm{CA}$ and $\mathrm{DB}$ are perpendicular to $\mathrm{AB}$.
Yes CA and DB are parallel.

Construction:
(i) Drawn a line segment $\mathrm{AB}$ of length $6 \mathrm{~cm}$.
(ii) Place the set square on the line in such a way that the vertex of its right angle coincides with B first and $\mathrm{A}$ next and one arm of the right angle coincides with the line $\mathrm{AB}$.
(iii) Drawn lines $\mathrm{DB}$ and $\mathrm{CA}$ through $\mathrm{B}$ and $\mathrm{A}$, the other arm of the right angle of the set square.
(iv) The line $\mathrm{CA}$ and $\mathrm{DB}$ are perpendicular to $\mathrm{AB}$ at $\mathrm{A}$ and $\mathrm{B}$.

Challenge Problems
Question $7 .$
Is a triangle possible with the angles $90^{\circ}, 90^{\circ}$ and $0^{\circ}$ ? Why?
Solution:
No, a triangle cannot have more than one right angle

Question $8 .$

Which of the following statements is true. Why?
(a) Every equilateral triangle is an isosceles triangle.
(b) Every isosceles triangle is an equilateral triangle.
Solution:
(a) It is true
In an equilateral triangle, all three sides are equal.
It can be an isosceles triangle also, which has two sides equal.
(b) But every isosceles triangle need not be an equilateral triangle.

Question 9 .
If one angle of an isosceles triangle is $70^{\circ}$, then find the possibilities for the other two angles.
Solution:
$70^{\circ}, 40^{\circ}$ (or) $55^{\circ}, 55^{\circ}$
 

Question $10 .$
Which of the following can be the sides of an isosceles triangle?
(a) $6 \mathrm{~cm}, 3 \mathrm{~cm}, 3 \mathrm{~cm}$
(b) $5 \mathrm{~cm}, 2 \mathrm{~cm}, 2 \mathrm{~cm}$

(c) $6 \mathrm{~cm}, 6 \mathrm{~cm}, 7 \mathrm{~cm}$
(d) $4 \mathrm{~cm}, 4 \mathrm{~cm}, 8 \mathrm{~cm}$
Solution:
In a triangle sum of any two sides greater than the third side (a), (b) and (d) cannot form a triangle.
(c) can be the sides of an isosceles triangle.

Question 11.
Study the given figure and identify the following triangles.

(a) equilateral triangle
(b) isosceles triangles
(c) scalene triangles
(d) acute triangles
(e) obtuse triangles
(f) right triangles

Solution:
(a) $\mathrm{BC}=1+1+1+1=4 \mathrm{~cm}$
$\mathrm{AB}=\mathrm{AC}=4 \mathrm{~cm}$
$\triangle \mathrm{ABC}$ is an equilateral triangle.
(b) $\triangle \mathrm{ABC}$ and $\triangle \mathrm{AEF}$ are isosceles triangles.
Since $\mathrm{AB}=\mathrm{AC}=4 \mathrm{~cm} \mathrm{Also} \mathrm{AE}=\mathrm{AF}$.
(c) In a scalene triangle, no two sides are equal.
$\triangle \mathrm{AEB}, \triangle \mathrm{AED}, \triangle \mathrm{ADF}, \triangle \mathrm{AFC}, \triangle \mathrm{ABD}, \triangle \mathrm{ADC}, \Delta \mathrm{ABF}$ and $\triangle \mathrm{AEC}$ are scalene triangles.
(d) In an acute-angled triangle all the three angles are less than $90^{\circ}$.
$\triangle \mathrm{ABC}, \Delta \mathrm{AEF}, \Delta \mathrm{ABF}$ and $\triangle \mathrm{AEC}$ are acute-angled triangles.
(e) In an obtuse-angled triangle any one of the angles is greater than $90^{\circ}$.
$\triangle \mathrm{AEB}$ and $\triangle \mathrm{AFC}$ are obtuse angled triangles.
(f) In a right triangle, one of the angles is $90^{\circ}$.
$\triangle \mathrm{ADB}, \triangle \mathrm{ADC}, \triangle \mathrm{ADE}$ and $\triangle \mathrm{ADF}$ are right-angled triangles.
 

Question $12 .$
Two sides of the triangle are given in the table. Find the third side of the triangle?

Solution:

Question 13..
Complete the following table.

Solution:

(i) Always acute angles
(ii) Acute angle
(iii) Obtuse angle