3 Marks 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^\circ , 60^\circ $and $40^\circ .$

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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^\circ $
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^\circ , 60^\circ $ and $100^\circ .$

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Question 33 Marks

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

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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 $\angle\text{A}+\angle\text{A}=\angle\text{A}+\angle\text{B}+\angle\text{C}$
$\Rightarrow2\angle\text{A}=180^\circ$^\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^\circ $. Find the other acute angle.

Answer

Sum of three angles of a right triangle $= 180^\circ $
Sum of two acute angles $= 180^\circ - 90^\circ = 90^\circ $
Measure of one angle $= 50^\circ $
Second acute angle $= 90^\circ - 50^\circ = 40^\circ $

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