Question

find the value of x ​

Answer 1

Answer:

Value of x is 105°.

Step-by-step explanation:

Given :

• ∠POR =35°.
• ∠OQS = 40°.
• PO = OQ.
• RO = OS.

To find :

• Value of x.

Solution :

In ∆POR and ∆SOQ :

✧ PO = OQ [Given]

✧ RO = OS [Given]

✧ ∠POR = ∠SOQ = 35° [By vertically opposite angle property which means when two lines are intersecting at any point then vertically opposite angles are equal].

By SAS (Side Angle Side) congruence rule :

∆POR is congruent to ∆SOQ.

By CPCT

∠RPO = ∠OQS = 40° ----(1)

Now,

We know,

✿ Sum of all interior angles of triangle is equal to 180°. This property is also known as "Angle sum property".

So,

$\implies$ ∠RPO + ∠POR + x = 180°

• Put ∠RPO = 40° [By equation (1)] and ∠POR = 35°.

$\implies$ 40° + 35° + x = 180°

$\implies$ 75° + x = 180°

$\implies$ x = 180° - 75°

$\implies$ x = 105°

Therefore,

Value of x is 105°.

Answer 2

Given -

• ∠POR = 35°
• ∠OQS = 40°
• PO = QO
• OR = OS

To find -

• the value of x, i.e, ∠POR

Solution -

We can observe that the two lines PQ and RS are interesting at a point O.

And also we can spot two triangles in the figure formed by the two lines.

Therefore to find the answer to your question we have to prove the given triangles congruent.

In △POR and △QOS,

S { PO = QO (given)

A { ∠POR = ∠QOS = 35° (vertically opposite angles)

S { OR = OS (given)

△POR ≅△QOS (by SAS Rule)

By CPCT,

∠PRO = ∠QSO

Further,

∠POR = ∠QOS = 35°

In △QOS,

=> ∠OQS + ∠QOS + ∠QSO = 180° (Sum of all sides of a triangle equals to 180°)

=> 40° + 35° + ∠QSO = 180°

=> 75° + ∠QSO = 180°

=> ∠QSO = 180° - 75°

=> ∠QSO = 105°

As stated above,

∠PRO = ∠QSO = 105°

Hence, the value of x = 175°

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Briefing the terms mentioned above -

The SAS (Side Angle Side) rule states that, if two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

Intersecting lines are lines that cross each other. The point where they meet is called a vertex. When two lines intersect, the opposite angles are equal. These angles are called vertically opposite angles because they are opposite each other at a vertex.

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