Question

The degree measure of each of the three angles of a triangle is an integer. Which of the following could not be the ratio of their measures?

• A. $2:3:4$
• B. $3:4:5$
• C. $5:6:7$
• D. $6:7:8$

Answer 1

The correct option is D

$6:7:8$

Explanation for the correct option:

Correct option: (D)

Find the ratio of their measures:

Let us assume $6x,7x$ and $8x$ be the three angles,

Then $6x+7x+8x=180°$

$21\mathrm{x}=180°$

$x=\frac{180°}{21}$

Then,

$$\begin{gathered} 6x=6\times \frac{180°}{21} \\ =51.428 \end{gathered}$$,

which is not an integer.

Explain the incorrect option:

(A)Let us assume $2x,3x$ and $4x$ be the three angles,

Then $2x+3x+4x=180°$ (Sum of all the angles of triangle ${180}^o$)

$$\begin{gathered} \Rightarrow 9\mathrm{x}={180}^o \\ \Rightarrow x={20}^o \\ \therefore x={20}^o \end{gathered}$$

Then,

$$\begin{gathered} 3x=3\times {15}^o \\ ={45}^o \end{gathered}$$,

which is an integer.

(B)Let us assume $3x,4x$ and $5x$ be the three angles,

Then $3x+4x+5x=180°$ (Sum of all the angles of triangle ${180}^o$)

$$\begin{gathered} \Rightarrow 12\mathrm{x}={180}^o \\ \Rightarrow x={15}^o \\ \therefore x={15}^o \end{gathered}$$

Then,

$$\begin{gathered} 2x=2\times {20}^o \\ ={40}^o \end{gathered}$$,

which is an integer.

(C)Let us assume $5x,6x$ and $7x$ be the three angles,

Then $5x+6x+7x=180°$ (Sum of all the angles of triangle ${180}^o$)

$$\begin{gathered} \Rightarrow 18\mathrm{x}={180}^o \\ \Rightarrow x={10}^o \\ \therefore x={10}^o \end{gathered}$$

Then,

$$\begin{gathered} 5x=5\times {10}^o \\ ={50}^o \end{gathered}$$,

which is an integer.

Hence, option (D) is correct answer.