3 Mark Question

Question 13 Marks

In a $\triangle\text{ABC},$ if $3\angle\text{A}=4\angle\text{B}=6\angle\text{C},$ calculate the angles.

Answer

In a $\triangle\text{ABC}$
$3\angle\text{A}=4\angle\text{B}=6\angle\text{C}=1$ (say)
$\therefore\angle\text{A}=\frac{1}{3}$
$\angle\text{B}=\frac{1}{4}$
$\angle\text{C}=\frac{1}{6}$
$\therefore$ Ratio $=\frac{1}{3}:\frac{1}{4}:\frac{1}{6}=\frac{4:3:2}{12}$
(LCM of 3, 4, 6 = 12)
Sum of angles $\triangle\text{ABC}=180^\circ$
$\therefore\angle\text{A}=\frac{180^\circ\times4}{4+3+2}=\frac{180^\circ\times4}{9}=80^\circ$
$\angle\text{B}=\frac{180^\circ\times3}{9}=60^\circ$
$\angle\text{C}=\frac{180^\circ\times2}{9}=40^\circ$
Hence, angles of $\triangle\text{ABC}$ are 40°, 60°and 40°.

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Question 23 Marks

The angles of a triangle are in the ratio 1 : 3 : 5. Find the measure of each one of the angles.

Answer

Sum of three angles of a triangle = 180°
Ratio of three angles = 1 : 3 : 5
$\therefore\text{First angle}=\frac{180^\circ\times1}{1+3+5}=\frac{180^\circ\times1}{9}=20^\circ$
$\text{Second angle}=\frac{180^\circ\times3}{9}=60^\circ$
$\text{Third angle}=\frac{180^\circ\times5}{9}=100^\circ$
Hence, three angles are 20°, 60° and 100°.

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Question 33 Marks

One of the angles of a triangle is 110° and the other two angles are equal. What is the measure of each of these equal angles?

Answer

Let the measure of each of the equal angles be x°. Then,
x° + x° + 110° = 180°
(Angle sum property of a triangle)
⇒ 2x° + 110° = 180°
⇒ 2x° = 180° - 110° = 70°
$\Rightarrow\text{x}^\circ=\Big(\frac{70}{2}\Big)^\circ=35^\circ$
The measure of each of the equal angles is 35°.

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Question 43 Marks

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

Answer

Let the three angles of a triangle be$\angle\text{A},\angle\text{B},\angle\text{C}.$Then,$\angle\text{A}=\angle\text{B}+\angle\text{C}$
Adding $\angle\text{A}$to both sides, we get $\angle\text{A}+\angle\text{A}=\angle\text{A}+\angle\text{B}+\angle\text{C}$
$\Rightarrow2\angle\text{A}=180^\circ$°
(Angle sum property of a triangle)
$\Rightarrow\angle\text{A}=\Big(\frac{180^\circ}{2}\Big)=90^\circ$
One of the angles of the triangle is a right angle.
Hence, the triangle is a right triangle.

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Question 53 Marks

One of the acute angles of a right triangle is 50°. Find the other acute angle.

Answer

Sum of three angles of a right triangle = 180°
Sum of two acute angles = 180° - 90° = 90°
Measure of one angle = 50°
Second acute angle = 90° - 50° = 40°

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