Question

In Fig, line l ∥ m and a transversal n cuts them P and Q respectively. If ∠1 = 75°, find all other angles.
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Answer 1

Given :-

• ∠1 = 75°

Solution :-

• To calculate all the angles we have to use some properties of angles. The properties of angles are alternate , corresponding, vertically opposite angles are used to find the angles.

Explanation :-

∠1 = ∠3 (Vertically opposite angles)

∠4 = ∠2 (Vertically opposite angles)

∠5 = ∠7 (Vertically opposite angles)

∠6 = ∠8 (Vertically opposite angles)

Let's find ∠2 by using linear pair :-

∠1 + ∠2 = 180

75 + ∠2 = 180

∠2 = 180 - 75

∠2 = 105°

Therefore, ∠1 = ∠3 = 75° (Vertically opposite angles)

Therefore, ∠2 = ∠4 = 105° (Vertically opposite angles)

Let's find ∠6 by using corresponding :-

∠6 = ∠2 = 105° (corresponding angles)

∠6 = ∠8 = 105° (Vertically opposite angles)

Therefore, ∠6 = ∠8 = 105° (Vertically opposite angles)

Let's find ∠5 by using linear pair :-

∠5 + ∠6 = 180

∠5 + 105 = 180

∠5 = 180 - 105

∠5 = 75

Therefore, ∠5 = ∠7 = 75° (Vertically opposite angles)

Hence, the required angles are :-

• ∠1 = ∠3 = 75°
• ∠4 = ∠2 = 105°
• ∠5 = ∠7 = 75°
• ∠6 = ∠8 = 105°

Answer 2

Question:

• In given Fig, line l ∥ m and a transversal n cuts them at points P and Q respectively. If ∠1 = 75°, Find all other angles.

Answer:

★ Measure of other angles ::

• ∠2 = 105°
• ∠3 = 75°
• ∠4 = 105°
• ∠5 = 75°
• ∠6 = 105°
• ∠7 = 75°
• ∠8 = 105°

Explanation:

Given that:

• ⚘ ∠1 = 75°

To Find:

• ⚘ Measure of other angles?

Solution:

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➨ ∠3 = ∠1 (Vertically opposite angles)

➨ ∠3 = 75° $\tt \Big\lgroup \because\:\angle{1} = 75^{\circ} \Big\rgroup$

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➨ ∠7 = ∠3 (Corresponding angles)

➨ ∠7 = 75° $\tt \Big\lgroup \because\:\angle{3} = 75^{\circ} \Big\rgroup$

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➨ ∠5 = ∠7 (Vertically opposite angles)

➨ ∠5 = 75° $\tt \Big\lgroup \because\:\angle{7} = 75^{\circ} \Big\rgroup$

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➨ ∠5 + ∠8 = 180° (Linear pair)

➨ 75° + ∠8 = 180° $\tt \Big\lgroup \because\:\angle{5} = 75^{\circ} \Big\rgroup$

➨ ∠8 = 180° - 75°

➨ ∠8 = 105°

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➨ ∠4 = ∠8 (Corresponding angles)

➨ ∠4 = 105° $\tt \Big\lgroup \because\:\angle{8} = 105^{\circ} \Big\rgroup$

━━━━━━━━━━━━━━━━━━━━━━━━━

➨ ∠2 = ∠4 (Vertically opposite angles)

➨ ∠2 = 105° $\tt \Big\lgroup \because\:\angle{4} = 105^{\circ} \Big\rgroup$

━━━━━━━━━━━━━━━━━━━━━━━━━

➨ ∠6 = ∠2 (Corresponding angles)

➨ ∠6 = 105° $\tt \Big\lgroup \because\:\angle{4} = 105^{\circ} \Big\rgroup$

━━━━━━━━━━━━━━━━━━━━━━━━━

Therefore,

★ Required angles ::

• ∠2 = 105°
• ∠3 = 75°
• ∠4 = 105°
• ∠5 = 75°
• ∠6 = 105°
• ∠7 = 75°
• ∠8 = 105°

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Know More:

• Angles that are opposite to each other at a vertex are called vertically opposite angles. There measure is always equal.

• The angles which are formed in corresponding corners with the transversal cutting two parallel lines are called corresponding angles. There measure is always equal.

• Pair of adjacent angles, which is formed when two lines intersect is called linear pair. Sum of that two angles is 180°.

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