Question

The figure shows a parallelogram ABCD. The bisectors of angles A,B,C,D enclose a quadrilateral PQRS. Using the exterior angle theorem prove that each angle of quadrilateral PQRS is a right angle.

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Answer 1

Step-by-step explanation:

Given :-

The figure shows a parallelogram ABCD. The bisectors of angles A,B,C,D enclose a quadrilateral PQRS.

To find :-

Using the exterior angle theorem prove that each angle of quadrilateral PQRS is a right angle.

Solution :-

Given that

ABCD is a Parallelogram

The bisectors of angles A,B,C,D enclose are AP, BR ,CS ,DQ respectively enclose a quadrilateral PQRS

We know that

The adjacent angles are supplementary in a Parallelogram

=> <A+ <D = 180°

On dividing by 2 both sides then

=> <A/2 + <D/2 = 180°/2

=> <PAD + < PDA = 90° ---------(1)

Now,

In ∆ DPA , AP is extended to S then

The exterior angle DPS is equal to the sum of opposite two interior angles

=> < DPS = <PAD + < PDA

=> < DPS = 90° (from (1))

=> < DPS = <SPQ = 90° -------(2)

Since Vertically opposite angles are equal.

Similarly we can find

< PQR = 90° -----------(3)

<QRS = 90° ----------(4)

<RSP = 90°----------(5)

From (2),(3),(4)&(5)

Each angle in PQRS quadrilateral is equal to 90°

Hence, Proved.

Used formulae:-

• The adjacent angles are supplementary in a Parallelogram
• The exterior angle is equal to the sum of opposite two interior angles
• Vertically opposite angles are equal.

Answer 2

Answer:

To prove :-

• Using exterior angle theorem prove that each angle of quadrilateral PQRS is a right angle .

Proof:-

• Here given that

• ABCD is a parallelogram

• The bisectors of right angle ABCD encloses are AP,BR , CS,DQ respectively. encloses a quadrilateral PQRS.

• We know that,

• The adjacent angles are supplimentary in a Parallelogram.
• <A+<D=180degree.

• On dividing both sides we get,

• <PAD+<PDA=90degree

1. The exterior angles DPS is equal to the sum of opposite two interior angles.

• <DPS=<PAD+<PDA
• <DPS= 90 degree (ii)

• We know that vertical opposite angles are equal.

1. <PQR=90degree (3)
2. <QRS=90degree (4)
3. <RSP=90degree (5)

• From 2,3,4,5

1. Each angle in PQRS quadrilateral is equal to 90 degree.

Hope it helps u mate.

Answered by Rohith kumar maths dude