RS Aggarwal 2019,2020 solution class 7 chapter 15 Property of Traingles Exercise 15A

Exercise 15A

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Question 1:

In a ∆ABC, if ∠A = 72° and ∠B = 63°, find ∠C.

Answer 1:

Sum of the angles of a triangle is 180°.

$$\begin{gathered} \therefore \angle A+\angle B+\angle C = 180° \\ 72° + 63° +\angle C = 180° \\ \angle C = 45° \end{gathered}$$

Hence, ∠C measures 45°.

Question 2:

In a ∆DEF, if ∠E = 105° and ∠F = 40°, find ∠D.

Answer 2:

Sum of the angles of any triangle is 180°.
In ∆DEF:
$$\begin{gathered} \angle \mathrm{D}+\angle \mathrm{E}+\angle \mathrm{F}=180° \\ \angle \mathrm{D}+105°+40°=180° \\ \mathrm{or} \angle \mathrm{D}=180°-(105°+40°) \\ \mathrm{or} \angle \mathrm{D}=35° \end{gathered}$$

Question 3:

In a ∆XYZ, if ∠X = 90° and ∠Z = 48°, find ∠Y.

Answer 3:

Sum of the angles of any triangle is 180°.
In ∆XYZ:

$$\begin{gathered} \angle X+\angle Y+\angle Z=180° \\ 90°+\angle Y+48°=180° \\ =>\angle Y=180°-138°=42° \end{gathered}$$ 

Question 4:

Find the angles of a triangle which are in the ratio 4 : 3 : 2.

Answer 4:

Suppose the angles of the triangle are (4x)^o, (3x)^o and (2x)^o.

Sum of the angles of any triangle is 180^o.

∴ 4x + 3x + 2x = 180
9x = 180
x = 20
$\mathrm{Therefore}, \mathrm{the} \mathrm{angles} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{are} (4\times 20)°, (3\times 20)° \mathrm{and} ( 2\times 20)°, \mathrm{i}.\mathrm{e} . 80°, 60° \mathrm{and} 40°.$

Question 5:

One of the acute angles of a right triangle is 36°. find the other.

Answer 5:

Sum of the angles of a triangle is 180°.
Suppose the other angle measures x.

It is a right angle triangle. Hence, one of the angle is 90°.

$$\begin{gathered} \therefore 36° + 90° +x = 180° \\ x= 54° \end{gathered}$$
Hence, the other angle measures 54°.

Question 6:

The acute angles of a right triangle are in the ratio 2 : 1. Find each of these angles.

Answer 6:

Suppose the acute angles are (2x)^° and (x)^°
Sum of the angles of any triangle is 180°

∴ 2x+x+ 90$= 180$
$\Rightarrow$(3x) = 180-90
$\Rightarrow$(3x) = 90
$\Rightarrow$ x = 30
$$\begin{gathered} \mathrm{So}, \mathrm{the} \mathrm{angles} \mathrm{measure} (2\times 30)° \mathrm{and} 30° \\ i.e. 60° \mathrm{and} 30° \end{gathered}$$

Question 7:

One of the angles of a triangle is 100° and the other two angles are equal. Find each of the equal angles.

Answer 7:

The other two angles are equal. Let one of these angles be x°.

Sum of angles of any triangle is 180°.

∴ x + x+ 100 = 180
2x = 80
x = 40

Hence, the equal angles of the triangle are 40° each.

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Question 8:

Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.

Answer 8:

Suppose the third angle of the isosceles triangle is x^o.
Then, the two equal angles are (2x)^o and (2x)^o.
Sum of the angles of any triangle is 180^o.

∴ 2x +2x+ x= 180
5x  = 180
x = 36

Hence, the angles of the triangle are $36°, (2\times 36)° \mathrm{and} (2\times 36)°, \mathrm{i}.\mathrm{e}. 36°, 72°\mathrm{and} 72°$.

Question 9:

If one angle of a triangle is equal to the sum of the other two, show that the triangle is right-angled.

Answer 9:

$$\begin{gathered} \mathrm{Suppose} \mathrm{the} \mathrm{angles} are \angle A, \angle B and\angle C. \\ Given: \\ \angle A = \angle B +\angle C \\ Also, \angle A +\angle B+\angle C = 180° \\ \therefore \angle A+\angle A= 180° \\ \Rightarrow 2\angle A = 180° \\ \Rightarrow \angle A=90° \end{gathered}$$                  (Sum of the angles of a triangle is 180^°)

Hence, the triangle ABC is right angled at $\angle$A.

Question 10:

In a ∆ABC, if 2∠A = 3∠B = 6∠C, calculate ∠A, ∠B and ∠C.

Answer 10:

Suppose: 2∠A = 3∠B = 6∠C = x^°
Then, ∠A =  ${\left(\frac{x}{2}\right)}^{°}$
$\angle B ={\left(\frac{x}{3}\right)}^{°}\mathrm{and} \angle C =\left({\displaystyle \frac{x}{6}}\right)$^°

Sum of the angles of any triangle is 180°.
∠A +∠B +∠C = 180$°$
$$\begin{gathered} \Rightarrow \frac{x}{2}+\frac{x}{3}+\frac{x}{6}= 180° \\ \Rightarrow \frac{3x+2x+x}{6}=180° \\ \Rightarrow \frac{6x}{6}=180° \\ \Rightarrow x=180 \end{gathered}$$


 $\therefore \angle A={\left(\frac{180}{2}\right)}^{°}=90°$
$$\begin{gathered} \angle B ={\left(\frac{180}{3}\right)}^{°}={60}^{°} \\ \angle C ={\left(\frac{180}{6}\right)}^{°}=30° \end{gathered}$$

Question 11:

What is the measure of each angle of an equilateral triangle?

Answer 11:

We know that the angles of an equilateral triangle are equal.
Let the measure of each angle of an equilateral triangle be x^°.

∴ x + x + x = 180
x = $60$
Hence, the measure of each angle of an equilateral triangle is $60°$.

Question 12:

In the given figure, DE || BC. If ∠A = 65° and ∠B° = 55, find

(i) ∠ADE
(ii) ∠AED
(iii) ∠C

Answer 12:

(i)
$$\begin{gathered} DE\parallel BC \\ \therefore \angle ABC =\angle ADE = 55° \end{gathered}$$                   
(Corresponding angles)

(ii) Sum of the angles of any triangle is 180°.

$$\begin{gathered} \therefore \angle A+\angle B+\angle C = 180° \\ \angle C = 180° -(65° +55°) = 60° \end{gathered}$$

 DE || BC
$\therefore \angle AED =\angle ACB =60°$                (corresponding angles)

(iii)  We have found in point (ii) that $\angle C \mathrm{is} \mathrm{equal} \mathrm{to} 60°.$

Question 13:

Can a triangle have

(i) two right angles?
(ii) two obtuse angles?
(iii) two acute angles?
(iv) all angles more than 60°?
(v) all angles less than 60°?
(vi) al angles equal to 60°?

Answer 13:

(i) No. This is because the sum of all the angles is 180°.
(ii) No. This is because a triangle can only have one obtuse angle.
(iii) Yes
(iv) No. This is because the sum of the angles cannot be more than 180°.
(v) No. This is because one angle has to be more than 60° as the sum of all angles is always 180°.
(vi) Yes, it will be an equilateral triangle.

Question 14:

Answer the following in 'Yes' or 'No'.

(i) Can an isosceles triangle be a right triangle?
(ii) Can a right triangle be a scalene triangle?
(iii) Can a right triangle be an equilateral triangle?
(iv) Can an obtuse triangle be an isosceles triangle?

Answer 14:

(i) Yes, it will be an isosceles right triangle.

(ii) Yes, a right triangle can have all sides of different measures. For example, 3, 4 and 5 are the sides of a scalene right triangle.

(iii) No, it cannot be an equilateral triangle since the hypotenuse square will be the sum of the square of the other two sides.

(iii) Yes, if an obtuse triangle has an obtuse angle of 120° and the other two angles of 30° each, then it will be an isosceles triangle.

Question 15:

Fill in the blanks:

(i) A right triangle cannot have an ...... angle.
(ii) The acute angles of a right triangle are ...... .
(iii) Each acute angle of an isosceles right triangle measures ...... .
(iv) Each angle of an equilateral triangle measures ...... .
(v) The side opposite the right angle of a right triangle is called ...... .
(vi) The sum of the lengths of the sides of a triangle is called its...... .

Answer 15:

(i) obtuse (since the sum of the other two angles of the right triangle is 90^o)
(ii) equal to the sum of 90^o
(iii) 45^o (since their sum is equal to 90^o)
(iv) 60^o
(v) a hypotenuse
(vi) perimeter