Question

In Fig. 6.44, the side QR of ∠PQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that
∠ZQTR = ∠ZQPR.
​

Answer 1

$\purple{\boxed{\bf{Given࿐}}}$

Bisector of ∠PQR and ∠PRS meet at point T

$\red{\boxed{\bf{тσ \:fเи∂}}}$

$\orange{\boxed{\bf{∠QTR = \frac{1}{2} ∠QPR}}}$

In ΔQTR, ∠TRS is an exterior angle.

∠QTR + ∠TQR = ∠TRS

∠QTR = ∠TRS − ∠TQR (1)

For ΔPQR, ∠PRS is an external angle.

∠QPR + ∠PQR = ∠PRS

∠QPR + 2∠TQR = 2∠TRS (As QT and RT are angle bisectors)

∠QPR = 2(∠TRS − ∠TQR)

∠QPR = 2∠QTR [By using equation (1)]

$\red{\boxed{\tt{∠QTR = ∠QPR}}}$

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$\red&#9998{\tt{Answer}}\: \orange☟$

⠀⠀$&#9679\purple{\bf{See \:the \:attachment}}\red{⇑}$

⠀ $&#9679\orange{\bf{Question\: solved}}\: \green✔$

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All necessary formulas⤵️

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$\purple{\boxed{\bf{Angle\:sum\: property}}}$

Angle sum property of triangle states that the sum of interior angles of a triangle is 180°. Proof .Thus, the sum of the interior angles of a triangle is 180°.

$\blue{\tt{Example:-}}$

$\red{\boxed{a+b+c=180°}}$

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$\orange{\boxed{\bf{Alternate\: interior\:angle}}}$

Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal.Alternate interior angles are equal if the lines intersected by the transversal are parallel.

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$\orange\star{\bf{\red{\underbrace{complementary \:angle}}}}\red\star$

The sum of 2 numbers$=90°$

example $a−b=90°$

how to find "a" if a is not mentioned

$\red{\underbrace{\bf{\orange{Given࿐}}}}$

$a= \: ?$

$b = 40$

$a+40=\:90°$

$a=90-40°$

$a=50°$

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$\pink\star{\bf{\purple{\underbrace{supplementary\: angle}}}}\red\star$

The sum of two numbers$= \:180°$

example $a+b=180°$

how to find "a" if a is not mentioned

Given

$a= \:?$

$b =\: 40$

$a+40=180°$

$a=180-40°$

$a=140°$

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$\orange\star{\bf{\green{\underbrace{Adjacent \:angle}}}}\red\star$

If there is a common ray between ${\bf&#x2220}a$ and ${\bf&#x2220}b$ so it is a adjacent angle.

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$\orange\star{\bf{\blue{\underbrace{Vertical\: opposite\: angle }}}}\red\star$

Vertical angles are pair angles formed when two lines intersect. Vertical angles are sometimes referred to as vertically opposite angles because the angles are opposite to each other.

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$\orange\star{\bf{\orange{\underbrace{lenear\: pair \:of\: angles}}}}\red\star$

Here ${\bf&#x2220}a$$+{\bf&#x2220}b$$=$180°.

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$\red{\bf{✿┅═══❁✿ Be\: Brainly✿❁═══┅✿}}$

Answer 2

Given࿐

Bisector of ∠PQR and ∠PRS meet at point T

∠QTR=

2

1

∠QPR

In ΔQTR, ∠TRS is an exterior angle.

∠QTR + ∠TQR = ∠TRS

∠QTR = ∠TRS − ∠TQR (1)

For ΔPQR, ∠PRS is an external angle.

∠QPR + ∠PQR = ∠PRS

∠QPR + 2∠TQR = 2∠TRS (As QT and RT are angle bisectors)

∠QPR = 2(∠TRS − ∠TQR)

∠QPR = 2∠QTR [By using equation (1)]

∠QTR=∠QPR\red{\boxed{\tt{∠QTR = ∠QPR}}}

∠QTR=∠QPR

All necessary formulas

Angle sum property

Angle sum property of triangle states that the sum of interior angles of a triangle is 180°. Proof .Thus, the sum of the interior angles of a triangle is 180°.

Example:−

a+b+c=180°\red{\boxed{a+b+c=180°}}

a+b+c=180°

Alternate interior angle

Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal.Alternate interior angles are equal if the lines intersected by the transversal are parallel.

complementary angle

The sum of 2 numbers=90°=90°=90°

example

how to find "a" if a is not mentioned

Given

a=?a= \: ?a=?

b=40b = 40b=40

a+40=90°a+40=\:90°a+40=90°

a=90−40°a=90-40°a=90−40°

a=50°a=50°a=50°

supplementary angle

The sum of two numbers=180°= \:180°=180°

example a+b=180°a+b=180°a+b=180°

how to find "a" if a is not mentioned

Given

a=?a= \:?a=?

b=40b =\: 40b=40

a+40=180°a+40=180°a+40=180°

a=180−40°a=180-40°a=180−40°

a=140°a=140°a=140°

Adjacent angle

If there is a common ray between ∠a{\bf∠}a∠a and ∠b{\bf∠}b∠b so it is a adjacent angel

Vertical opposite angle

Vertical angles are pair angles formed when two lines intersect. Vertical angles are sometimes referred to as vertically opposite angles because the angles are opposite to each other.

lenear pair of angles

⋆

Here ∠a{\bf∠}a∠a +∠b+{\bf∠}b+∠b === 180°.