Name: Equilateral Triangles
Uploaded: 2019-07-19
Duration: 3 min 15 s
Description: Equilateral triangles have all three sides of equal length. Learn more about such a triangle through this important video having examples.

Question

In the figure, $A B C D E$ is pentagon with all sides of the same length and all angles of the same size. The sides $A B$ and $A E$ extended, meet the side CD extended at $P$ and $Q$ .

Are the sides of $A P$ and $A Q$ of $A P Q$ equal? Why?

Accepted Answer

Here, all angles of the pentagon are equal (a regular polygon).

$∴ ∠ A B C = ∠ A E D . . . (1)$

Since $(∠ P B C, ∠ A B C)$ and $(∠ A E D, ∠ Q E D)$ form a linear pair.

$\Rightarrow ∠ P B C = 180 - ∠ A B C$ .....( $∵$ Linear angles are supplementary)

$= 180 - ∠ A E D$ ... from $(1)$

And $∠ P B C = ∠ Q E D$ and $∠ B C D = ∠ E D C . . . . (2)$ ....( $∵ ∠ A E D$ and $∠ Q E D$ form linear pair)

Again $(∠ B C D, ∠ B C P)$ and $(∠ E D C, ∠ E D Q)$ form a linear pair.

$\Rightarrow ∠ B C P = 180 ° - ∠ B C D$ (Linear angles are supplementary)

$= 180 - ∠ E D C$ ...from (2)

$\Rightarrow ∠ B C P = ∠ E D Q$ ... $(3)$

According to property,

If one side of a triangle and angle at its ends are equal to one side of another triangle and the angles at its ends, then the third angles are also equal and the sides opposite equal angle are equal.

In $△ P B C$ and $△ Q E D,$

$∠ B C D = ∠ E D C$ and $∠ B C P = ∠ E D Q$

Also, $B C = D E$ (a regular polygon have all sides equal.)

Hence, $P C = D Q . . . (∵ ∠ P B C = ∠ D E Q)$ and

$P B = E Q (∵ ∠ P C B = ∠ E D Q)$

Hence, the sides are equal in $△ P B C$ and $△ Q E D$ .

Now, as $A B = A E$ and $A E = E Q$ ....(3)

Since $A B + B P = A P$ and $A E + E Q = A Q$

$\Rightarrow A P = A Q$ in $△ A P Q$ .

In the figure, $A B C D E$ is pentagon with all sides of the same length and all angles of the same size. The sides $A B$ and $A E$ extended, meet the side CD extended at $P$ and $Q$ .

Are the sides of $A P$ and $A Q$ of $A P Q$ equal? Why?

Important Questions on Equal Triangles

Learn and Prepare for any exam you want !

In the figure, ABCDE is pentagon with all sides of the same length and all angles of the same size. The sides AB and AE extended, meet the side CD extended at P and Q. Are the sides of AP and AQ of APQ equal? Why?

Important Questions on Equal Triangles

Learn and Prepare for any exam you want !

In the figure, $A B C D E$ is pentagon with all sides of the same length and all angles of the same size. The sides $A B$ and $A E$ extended, meet the side CD extended at $P$ and $Q$ .

Are the sides of $A P$ and $A Q$ of $A P Q$ equal? Why?