## RS Aggarwal Class 7 Math Fifteenth Chapter Properties of Triangles Exercise 15A Solution

## EXERCISE 15A

**(1) In a ****∆ABC, if ****∠A = 72 ^{o} and ∠B = 63^{o}, find ∠C.**

Solution: We know that the sum of the angles of a triangle is 180^{o}.

∠A + ∠B + ∠C = 180^{o}

or, 72^{o} + 63^{o} + ∠C = 180^{o}

or, ∠C = 180^{o} – 135^{o}

or, ∠C = 45^{o }

**(2) In a ****∆****DEF, if ∠E = 105 ^{o} and ∠F = 40^{o}, find ∠D.**

Solution: We know that the sum of the angles of a triangle is 180^{o}.

∠D + ∠E + ∠F = 180^{o}

or, ∠D + 105^{o} + 40^{o} = 180^{o}

or, ∠D = 180^{o} – 145^{o}

or, ∠D = 35^{o}

**(3) In a ****∆****XYZ, if ∠X = 90 ^{o }and ∠Z = 48^{o}, find ∠Y.**

Solution: We know that the sum of the angles of a triangle is 180^{o}.

∠X + ∠Y + ∠Z = 180^{o}

or, 90^{o }+ ∠Y + 48^{o} = 180^{o}

or, ∠Y = 180^{o} – 138^{o}

or, ∠Y = 42^{o}

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

Solution: Let the measures of the given angles of the triangle be (4x)^{o}, (3x)^{o} and (2x)^{o}. Then,

4x + 3x + 2x = 180

or, 9x = 180

or, x = 180/9

or, x = 20

Hence, the angles of the triangle are 80^{o}, 60^{o} and 40^{o}.

**(5) One of the acute angles of a right triangle is 36 ^{o}, find the other.**

Solution: Let the measure of unknown angle be x^{o}.

We know that the sum of the angles of a triangle is 180^{o}.

∴ x + 90 + 36 = 180

or, x = 180 – 126

or, x = 54

Hence, each unknown angle is 54^{o}.

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

Solution: Let the measure of each angles be (2x)^{o} and x^{o}.

We know that the sum of the angles of a triangle is 180^{o}.

∴ 2x + x + 90 = 180

or, 3x = 180 – 90

or, 3x = 90

or, x = 30.

Hence the required angles are (2 × 30)^{o} = 60^{o} and 30^{o}.

**(7) One of the angles of a triangle is 100 ^{o} and the other two angles are equal. Find each of the equal angles.**

Solution: Let the measure of each unknown angle be x^{o}.

We know that the sum of the angles of a triangle is 180^{o}.

∴ x + x + 100 = 180

or, 2x = 180 – 100

or, 2x = 80

or, x = 40

Hence, each unknown angle is 40^{o}.

**(8) Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles of the triangle.**

Solution: Let the third angle be x^{o}.

∴ 2x + 2x + x = 180

or, 5x = 180

or, x = 36

Hence, the third angle is 36^{o} and each others are (36 × 2) = 72^{o}.

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

Solution: Given, ∠A = ∠B + ∠C

∴ ∠A + ∠B + ∠C = 180

or, ∠A + ∠A = 180

or, 2∠A = 180

or, ∠A = 90

Hence, prove that the triangle is right-angled.

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

**(11) What is the measure of each angle of an equilateral triangle?**

Ans: Each angle of an equilateral triangle measure 60^{o}.

**(12) In the given figure, DE ∥ BC. If ∠A = 65 ^{o} and ∠B = 55^{o}, find (i) ∠ADE; (ii) ∠AED; ∠C.**

(i) Since, DE ∥ BC and ADB is the transversal, so

∠ADE = ∠ABC = 55^{o }(corresponding angles)

(iii) In ∆ABC, ∠A = 65^{o}, ∠B = 55^{o}

∴ ∠A + ∠B + ∠C = 180

or, 65 + 55 + ∠C = 180

or, ∠C = 180 – 120

or, ∠C = 60

(ii) Since, DE ∥ BC and ADB is the transversal, so

∠C = ∠AED = 60^{o} (corresponding angles)

**(13) Can a triangle have**

(i) Two right angles? = NO

(ii) Two obtuse angles? = No

(iii) Two acute angles? = Yes

(iv) All angles more than 60^{o}? = No

(v) All angles less than 60^{o}? = No

(vi) All angles equal to 60^{o}? = Yes

**(14) Answer the following in “Yes” or “No”.**

(i) Can a isosceles triangle be a right triangle? = Yes

(ii) Can a right triangle be a scalene triangle? = Yes

(iii) Can a right triangle be an equilateral triangle? = No

(iv) Can an obtuse triangle be an isosceles triangle? = Yes

**(15) Fill in the blanks:**

(i) A right triangle cannot have an ** obtuse** angle.

(ii) The acute angles of a right triangle are ** complementary**.

(iii) Each acute angle of an isosceles right triangle measures ** 45^{o}**.

(iv) Each angle of an equilateral triangle measures ** 60^{o}**.

(v) The side opposite the right triangle is called ** hypotenuse**.

(vi) The sum of the lengths of the side of a triangle is called its ** perimeter**.

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