Chapter 7, Problem 5PE

To determine

To proof: The triangles EFG and IHG are similarwith the help of given information.

Explanation of Solution

Given information:

  

  $\overline{EF}\parallel \overline{JK};\overline{JK}\parallel \overline{HI}$

Explaination:

Consider a figure, $EFGH$ as shown above;

  $\overline{EF}\parallel \overline{JK};\overline{JK}\parallel \overline{HI}$ ;

Now consider two triangles, $EFG$ and $1\text{HG}$

As the line $EF$ is parallel to the line $JK$ and the line $JK$ is parallel to the line $HI$ therefore,

  $\overline{EF}\parallel \overline{HI}$

As the line $E/$ intersects the parallel line $EF$ and $H$ therefore, their corresponding angles should be congruent with each other.

  $\angle 2\cong \angle 3\dots \dots (1)$

And, the angles $EGF$ and $HG/$ are vertically opposite angles,

  $\angle EGF\cong \angle HGI\cdots \dots (2)$

Because the two triangles, $EFG$ and $IHG$ have two congruent angles, $(1),$ and $(2),$ therefore, according to the Angle-Angle Similarity Postulate.

  $\text{Δ}EFG\sim \text{Δ}IHG$

Hence it is proved.