Question

Pls answer the question by looking at the following image attached with the question

Topic - Triangle and its properties ​

Answer 1

Answer:

(a) Value of d is 142°.

(b) Value of b is 45°.

Step-by-step explanation:

(a) We have,

• c = 57°

• a = c/3 = 57/3 [As c is 57°] = 19°

We know,

✿ If two parallel lines are intersected by the transversal then their alternate interior angles are equal. This property is also known as Alternate interior angle.

Here, Line PQ || UT and they both are intersected by transversal PT.

So, ∠PTU = ∠TPQ

$\implies$ c = a + b

$\implies$ 57° = 19° + b

$\implies$ b = 57° - 19°

$\implies$ b = 38°

Now,

We also know that,

✿ Sum of two interior angles formed on same side of the transversal when two parallel lines intersect, is equal to 180°. This property is also known as Co-interior angle.

PQ || RS and PR is transversal.

So,

$\implies$ ∠QPR + ∠PRS = 180°

$\implies$ b + d = 180°

$\implies$ 38° + d = 180°

$\implies$ d = 180° - 38°

$\implies$ d = 142°

Therefore,

Value of d is 142°.

(b) We have,

• c = 75°

• a = (2/5) c = 2/5 × 75° = 30° .

Similarly, as part (a) by Alternate interior angle property.

Here, PQ || UT and PT is transversal.

$\implies$ ∠PTU = ∠TPQ

$\implies$ c = a + b

$\implies$ 75° = 30° + b

$\implies$ b = 75° - 30°

$\implies$ b = 45°

Therefore,

Value of b is 45°.

Answer 2

Answer:

Question :-

$\bigstar$ In given figure PQ, RS, and UT are parallel lines.

To Find :

(a) If c = 57° and a = c/3, find the value of d.

(b) If c = 75° and a = 2/5c, find the value of b.

Solution :-

(a) :-

First, we have to find the value of a :

Given :

• c = 57°

So,

$\leadsto \bf a =\: \dfrac{c}{3}$

$\leadsto \sf a =\: \dfrac{57^{\circ}}{3}$

$\leadsto \sf a =\: \dfrac{19^{\circ}}{1}$

$\leadsto \sf\bold{\green{a =\: 19^{\circ}}}$

Now, we have :

$\dashrightarrow \bf PQ \parallel UT$

Hence,

$\bullet \: \: \sf \angle{c} =\: \angle{QPT} \: \: \bigg\lgroup \sf\bold{\pink{Alternate\: Angles}}\bigg\rgroup$

Also,

$\bullet \: \: \sf \angle{QPT} =\: a + b$

Then,

$\implies \bf c =\: a + b$

$\implies \sf 57^{\circ} =\: 19^{\circ} + b$

$\implies \sf 57^{\circ} - 19^{\circ} =\: b$

$\implies \sf 38^{\circ} =\: b$

$\implies \sf\bold{\orange{b =\: 38^{\circ}}}$

Again, we have :

$\dashrightarrow \bf PQ \parallel RS$

Now, as we know that :

$\clubsuit$ The sum of each interior angle, when the lines are parallel then the angle is equal to 180°.

So,

$\implies \bf b + d =\: 180^{\circ}$

$\implies \sf 38^{\circ} + d =\: 180^{\circ}$

$\implies \sf d =\: 180^{\circ} - 38^{\circ}$

$\implies \sf\bold{\red{d =\: 142^{\circ}}}$

Hence, the value of d is 142° .

$\sf\boxed{\bold{\purple{\therefore\: The\: value\: of\: d\: is\: 142^{\circ}\: .}}}$

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(b) :-

First, we have to find the value of a :

Given :

• c = 75°

So,

$\leadsto \bf a =\: \dfrac{2}{5}c$

$\leadsto \sf a =\: \dfrac{2}{5} \times 75^{\circ}$

$\leadsto \sf a =\: \dfrac{150^{\circ}}{5}$

$\leadsto \sf a =\: \dfrac{30^{\circ}}{1}$

$\leadsto \sf\bold{\green{a =\: 30^{\circ}}}$

Now, we have :

$\dashrightarrow \bf PQ \parallel UT$

Hence,

$\bullet \: \ \sf \angle{c} =\: \angle{QPT}\: \: \: \bigg\lgroup \sf\bold{\pink{Alternate\: Angle}}\bigg\rgroup$

Also,

$\bullet \: \: \sf \angle{QPT} =\: a + b$

Then,

$\implies \bf c =\: a + b$

$\implies \sf 75^{\circ} =\: 30^{\circ} + b$

$\implies \sf 75^{\circ} - 30^{\circ} =\: b$

$\implies \sf 45^{\circ} =\: b$

$\implies \sf\bold{\red{b =\: 45^{\circ}}}$

Hence, the value of b is 45° .

$\sf\boxed{\bold{\purple{\therefore\: The\: value\: of\: b\: is\: 45^{\circ}\: .}}}$

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