Question

find the unknowns in the following​

Answer 1

Answer:

The value of variables is x = 20° & y = 50°.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given that,

□ ROSE is a parallelogram.

m∠RES = ( 3x - 10 )°

m∠OSE = ( 5x + 30 )°

m∠ROS = y°

We have to find the values of the variables i. e. x and y.

Now,

We know that,

Opposite angles of a parallelogram are congruent.

∴ m∠RES = m∠ROS

⇒ ( 3x - 10 )° = y°

⇒ 3x - 10 = y

⇒ 3x - y = 10 - - ( 1 )

Now, we know that,

Adjacent angles of a parallelogram are supplementary.

∴ m∠RES + m∠OSE = 180°

⇒ ( 3x - 10 )° + ( 5x + 30 )° = 180°

⇒ 3x - 10 + 5x + 30 = 180

⇒ 3x + 5x - 10 + 30 = 180

⇒ 8x + 20 = 180

⇒ 8x = 180 - 20

⇒ 8x = 160

⇒ x = 160 ÷ 8

⇒ x = 20°

By substituting x = 20 in equation ( 1 ), we get,

3x - y = 10 - - ( 1 )

⇒ 3 ( 20 ) - y = 10

⇒ 60 - y = 10

⇒ - y = 10 - 60

⇒ - y = - 50

∴ y = 50° - - ( 2 )

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Verification:

We know that,

m∠RES = m∠ROS

Now,

m∠RES = ( 3x - 10 )°

⇒ m∠RES = 3 ( 20 ) - 10

⇒ m∠RES = 60 - 10

⇒ m∠RES = 50° - - ( 3 )

Now,

m∠ROS = y°

⇒ m∠ROS = 50° - - [ From ( 2 ) ]

∴ m∠RES = m∠ROS

Hence verified!

Also, we know that,

m∠RES + m∠OSE = 180°

LHS = m∠RES + m∠OSE

⇒ LHS = 50° + ( 5x + 30 )° - - [ From ( 3 ) ]

⇒ LHS = 50 + 5 ( 20 ) + 30

⇒ LHS = 50 + 100 + 30

⇒ LHS = 150 + 30

⇒ LHS = 180

RHS = 180

∴ LHS = RHS

Hence verified!

Answer 2

⭐WHAT TO KNOW TO SOLVE THIS QUESTION:

(1) Opposite angles of a parallelogram are congruent.

(2) Adjacent sides of a parallelogram are supplementary.

⭐Given:

• Quadrilateral ROSE is a parallelogram.

• <RES=(3x-10°)

• <OSE=(5x+30°)

• <SOR=y

• <ERO=x

⭐To Find:

• <SOR=y=?

• <ERO=x=?

⭐Solution:

Here, Quadrilateral ROSE is a parallelogram.

Since,

We know that,

• Adjacent Sides of a parallelogram are supplementary.

: . <RES+<OSE=180°

: . (3x-10°) +(5x+30°) =180°

: . 3x-10+5x+30=180°

: . 3x+5x-10+30=180°

: . 8x+20=180°

: . 8x=180°-20

: . 8x=160

: . x=160/8

: . x=20

Therefore, <RES=(3x-10) =3×20-10=50..... (1)

<OSE=(5x+30) =5×20+30=130........ (2)

Again We know that,

• Opposite sides of a parallelogram are congruent.

: . <RES=<ROS=y

: . <ROS=y=50°........ [From (1) ]

and <OSE=<ORE=x

: . <ORE=x=130°........ [From (2) ]

Therefore Values of x and y are 130° and 50°respectively.

⭐VERIFICATION

We know that,

• Sum of all angles of a quadrilateral is 360°.

: . <RES+<ESO+<SOR+<ORE=360°

: . L. H. S=50°+130°+50°+130°=RH.S

Hence Verified that L. H. S =R.H.S