Question

Question In Attachment⤴️
Step by step explaination needed
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Answer 1

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Given —

• EF || CD
• ∠ ABC = 40°
• ∠ CEF = 165°
• ∠ BCE = 24°

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To prove — AB || CD

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Construction — Draw EO perpendicular to CD.

• Hence, ∠ COE = 90° ( —1
• And, ∠ OEF = 90° ( —2

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Also, ∠ CEO + ∠ OEF = ∠ CEF

( Putting values, we get )

• Or, ∠ CEO + 90° = 165° ( From — 2 , Also, ∠ CEO And ∠ OEF are adjacent angles )
• Or, ∠ CEO = 165° — 90°
• Or, ∠ CEO = 75°

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Now, in triangle, CEO,

∠ CEO + ∠ COE + ∠ OCE = 180° { Angle sum property of a triangle }

Or, 75° + 90° + ∠ OCE = 180° { ∠ CEO = 75° — Proved above }

Or, ∠ OCE = 180° – 75° + 90°

Or, ∠ OCE = 180° – 165°

Or, ∠ OCE = 15 °

Or, ∠ DCE = 15 °

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Now, ∠ BCE + ∠ DCE = ∠ BCD

Or, 25° + 15° = ∠ BCD

Or, ∠ BCD = 40°

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Here, ∠ BCD = ∠ ABC = 40°

Also, ∠ BCD and ∠ ABC are alternate interior angles.

( Theorem — If the pair of alternate interior angles are equal then the lines are parallel. )

Hence, AB || CD — Proved

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Answer 2

#CHeatedfromabove

#CHeatedfromabove

________________________________

Given —

EF || CD

∠ ABC = 40°

∠ CEF = 165°

∠ BCE = 24°

________________________________

To prove — AB || CD

________________________________

Construction — Draw EO perpendicular to CD.

Hence, ∠ COE = 90° ( —1

And, ∠ OEF = 90° ( —2

— Also, ∠ CEO + ∠ OEF = ∠ CEF

( Putting values, we get )

Or, ∠ CEO + 90° = 165° ( From — 2 , Also, ∠ CEO And ∠ OEF are adjacent angles )

Or, ∠ CEO = 165° — 90°

Or, ∠ CEO = 75°

________________________________

— Now, in triangle, CEO,

∠ CEO + ∠ COE + ∠ OCE = 180° { Angle sum property of a triangle }

Or, 75° + 90° + ∠ OCE = 180° { ∠ CEO = 75° — Proved above )

Or, ∠ OCE = 180° – 75° + 90°

Or, ∠ OCE = 180° – 165°

Or, ∠ OCE = 15 °

Or, ∠ DCE = 15 °

________________________________

Now, ∠ BCE + ∠ DCE = ∠BCD

Or, 25° + 15° = ∠BCD

Or, ∠BCD = 40°

________________________________

Here, ∠BCD = ∠ABC = 40°

Also, ∠BCD and ∠ABC are alternate interior angles.

( Theorem — If the pair of alternate interior angles are equal then the lines are parallel. )

Hence, AB || CD — Proved

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#CHeatedfromabove