Yes, Euclid's fifth postulate imply the existence of parallel lines. Because if the two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other but if the sum of interior angles is 180, then the lines don't intersectRead more
Yes, Euclid’s fifth postulate imply the existence of parallel lines. Because if the two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other but if the sum of interior angles is 180, then the lines don’t intersect or lines are parallel.
In Fig. 6.40, ∠ X = 62°, ∠ XYZ = 54°. If YO and ZO are the bisectors of ∠ XYZ and ∠ XZY respectively of ∆ XYZ, find ∠ OZY and ∠ YOZ.
Given that : ∠X = 62° and ∠XYZ = 54° In △XYZ, ∠X + ∠XYZ + ∠XZY = 180° ⇒ 62° + 54° + ∠XZY = 180° ⇒ 116° + ∠XZY = 180° ⇒ ∠XZY = 180° - 116° = 64° YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively. therefore, ∠OYZ = 1/2 ∠XYZ = 1/2 × 54° = 27° ∠OYZ = 1/2 ∠XZY = 1/2 × 64° = 32° In △OYZ, ∠OZY + ∠ORead more
Given that : ∠X = 62° and ∠XYZ = 54°
See lessIn △XYZ, ∠X + ∠XYZ + ∠XZY = 180°
⇒ 62° + 54° + ∠XZY = 180°
⇒ 116° + ∠XZY = 180°
⇒ ∠XZY = 180° – 116° = 64°
YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively. therefore,
∠OYZ = 1/2 ∠XYZ = 1/2 × 54° = 27°
∠OYZ = 1/2 ∠XZY = 1/2 × 64° = 32°
In △OYZ, ∠OZY + ∠OYZ + ∠YOZ = 180°
⇒ 32° + 27° + ∠YOZ = 180°
⇒ 59° + ∠YOZ = 180°
⇒ ∠YOZ = 180° – 59° = 121°
In Fig. 6.41, if AB || DE, ∠ BAC = 35° and ∠ CDE = 53°, find ∠ DCE.
Given that : AB||DE, therefore ∠CED = ∠BAC [∵ Alternate Angles] ⇒ ∠CED = 35° In △CDE, ∠CED + ∠CDE + ∠DCE = 180° ⇒ 35° + 53° + ∠DCE = 180° ⇒ 88° = ∠DCE = 180° ⇒ ∠DCE = 180° - 88° = 92°
Given that : AB||DE, therefore
See less∠CED = ∠BAC [∵ Alternate Angles]
⇒ ∠CED = 35°
In △CDE, ∠CED + ∠CDE + ∠DCE = 180°
⇒ 35° + 53° + ∠DCE = 180°
⇒ 88° = ∠DCE = 180°
⇒ ∠DCE = 180° – 88° = 92°
In Fig. 6.14, lines XY and MN intersect at O. If ∠ POY = 90° and a : b = 2 : 3, find c.
Given: Lines XY and MN intersects at O, ∠POY = 90° and a:b = 2:3 Let, a = 2x, therefore b = 3x Here, ∠XOM + ∠POM + ∠POY = 180° [∵ XOY is a straight line] ⇒ 3x + 2x + 90° = 180° [∵ ∠POY = 90°] ⇒ 5x + 90° = 180° ⇒ 5x = 180° - 90° = 90° ⇒ x = 180°/5 = 18° Hence, ∠XOM = 3x = 3 × 18° = 54° and ∠POM = 2xRead more
Given: Lines XY and MN intersects at O,
See less∠POY = 90° and a:b = 2:3
Let, a = 2x, therefore b = 3x
Here, ∠XOM + ∠POM + ∠POY = 180° [∵ XOY is a straight line]
⇒ 3x + 2x + 90° = 180° [∵ ∠POY = 90°]
⇒ 5x + 90° = 180°
⇒ 5x = 180° – 90° = 90°
⇒ x = 180°/5 = 18°
Hence, ∠XOM = 3x = 3 × 18° = 54° and
∠POM = 2x = 2 × 18° = 36°
Here, ∠XON = ∠MOY
⇒ c = ∠POM + ∠POY [∵ Vertically Opposite Angles]
⇒ c = 36° + 90° = 126° [∵ ∠XON = c and ∠MOY = ∠POM + ∠POY ]
How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
If two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other.
If two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other.
See lessDoes Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Yes, Euclid's fifth postulate imply the existence of parallel lines. Because if the two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other but if the sum of interior angles is 180, then the lines don't intersectRead more
Yes, Euclid’s fifth postulate imply the existence of parallel lines. Because if the two lines intersect the third line in such a way that the sum of the interior angles is less than 180°, then the two lines intersect each other but if the sum of interior angles is 180, then the lines don’t intersect or lines are parallel.
See less