Question

The ______ of any two corresponding sides in two equiangular triangles is always the same.
A. Size
B. Limits
C. Measures
D. Ratio

Answer 1

Hint: We will take the help of the theorem that states that in two triangles, when the corresponding angles are equal, then their corresponding sides are in the same ratio. We will also take the help of similarity of triangles and using the AAA similarity in triangles, we will find the answer to the given question.

Complete step-by-step answer:
It is given in the question that we have to find in the blanks of the statement, the ______ of any two corresponding sides in two equiangular triangles is always the same. We know that in two triangles, if the corresponding angles are equal, then their corresponding sides are in the same ratio and they are similar triangles. This similarity is known as the AAA (Angle-Angle-Angle) similarity. We also know that angles of the equilateral triangle are equal and all of them are $60{}^{\circ }$. So, it means that their sides must be in the same ratio. For example, if we assume two triangles ABC and DEF as given in the figure below,

We can see that all the angles of triangle ABC and DEF are equal to $60{}^{\circ }$. Now, we will draw a line PQ in triangle DEF, such that $DP=AB$ and $DQ=AC$, and $\mathrm{\angle }DPQ=\mathrm{\angle }DQP=60{}^{\circ }$ as they both are the corresponding angles of $\mathrm{\angle }DEF,\mathrm{\angle }DFE$ respectively. Now, we will prove that $\mathrm{\Delta }ABC\cong \mathrm{\Delta }DPQ$. As we have assumed that the corresponding sides of $\mathrm{\Delta }ABC=\mathrm{\Delta }DPQ$, so it means that, $\mathrm{\angle }B=\mathrm{\angle }P=\mathrm{\angle }E$. We also know that PQ is parallel to EF, so it means that,
${\displaystyle \frac{DP}{PE}}={\displaystyle \frac{DQ}{QF}}$
Since, $DP=AB$ and $DQ=AC$, we also get,
${\displaystyle \frac{AB}{DE}}={\displaystyle \frac{AC}{DF}}$
Similarly, we get,
${\displaystyle \frac{AB}{DE}}={\displaystyle \frac{BC}{EF}}$
Therefore, we get,
${\displaystyle \frac{AB}{DE}}={\displaystyle \frac{AC}{DF}}={\displaystyle \frac{BC}{EF}}$
Hence, we have proved that triangle ABC is similar to triangle DEF, which means that the sides of the two similar triangles are in equal ratio. So, the ratio of any two corresponding sides in two equiangular triangles is always the same.
Therefore, the correct option is option D.

Note: Most of the students get confused with option A and option C. They think that if the angle measures are equal, then it means that by default they will have the equal measure of the corresponding sides too. It is not true because we can construct equilateral triangles of different measures, but the ratio of the sides of two equilateral triangles is the same. So, ratio is the correct answer.