Question

Multiple Choice Questions (MCQs)
9. The measure of x in the given figure is
(a) 82°
(b) 35°
(c) 63°
(d) 47°​

Answer 1

Note : For Understanding The Answer Very Clearly, We need Some Editing In Picture, The Edited Picture is in Attachment - 1 and Whole Answer is Designed Based on that Editing !! Thank you !

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Before Going into Answer, It's Important to know about some Basic Terms related to the Answer !!

• Angle : An angle is a combination of two rays with a common endpoint (Vertex) . An angle is represented by the symbol "∠"

• Alternate Interior Angles : Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal.

• Transversal Line : A Transversal is a line that crosses at least two other lines (usually parallel) lines

• Angle Sum Property : The sum of interior angles of triangles sums upto 180° according to Angle Sum Property

• Linear Pair : When The Measures Adjacent Angles adds upto 180°, Then the angles form Linear Pair. Linear pairs are always supplementary. The angles are said to be linear if they are adjacent to each other after the intersection of the two lines.

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\put(5,1){\vector(1,0){4}}\put(5,1){\vector(-1,0){4}}\put(5,1){\vector(1,1){3}}\put(2,3){ \underline{\boxed{\large\sf a + b = 180^{\circ}} }}\put(4.5,1.3){ \sf a^{\circ} }\put(5.7,1.3){ \sf b^{\circ} }\end{picture}$

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We need to find The measure of x in the given figure in :

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\put(0,0){\line(1,0){4}}\qbezier(0,0)(0,0)(-0.2,4.4)\qbezier(4,0)(4,0)(3.8,3.5)\put(-0.2,3.5){\line(1,0){4}}\qbezier(0,0)(0,0)(3.8,3.5)\put(2.3,0.01){\vector(1,0){0}}\put(2,3.5){\vector(1,0){0}}\multiput(-0.07,2)(-0.02,0.3){2}{\vector(0,1){0}}\multiput(3.89,2)(0,0.3){2}{\vector(0,1){0}}\qbezier(0.8,0)(1,0.35)(0.5,0.5)\qbezier(3.3,3)(3.5,2.7)(3.8,2.8)\qbezier(-0.2,3.9)(0.3,3.9)(0.4,3.5)\put(1,0.3){\bf X}\put(3.1,2.3){ \bf 35^{\circ} }\put(0.3,4){ \bf 82^{\circ} }\end{picture}$

✮ From the Attachment

• ∠EAB = 82°

• ∠CBD = 35°

• ∠BCD = x

• Given : ∠EAB = 82°

∠EAB + ∠BAC = 180°   (Linear Pair Property)

⇒ 82° + ∠BAC = 180°

⇒ ∠BAC = 180° - 82°

⇒ ∠BAC = 98°

• Given : ∠CBD = 35°

∠CBD = ∠ACB   (Alternate Interior Angles)

➠ ∠ACB = 35°

✧ Now We can Observe Triangle CAB

∠BAC + ∠ACB + ∠ABC = 180°   (Angle Sum Property)

⇒ 98° + 35° + ∠ABC = 180°

⇒ 133° + ∠ABC = 180°

⇒ ∠ABC = 180° - 133°

➠ ∠ABC = 47°

• So We have ∠ABC = 47°

∠ABC = x   (Alternate Interior Angles)

➠ x = 47°

Therefore

• Measure of x in the given figure = 47°

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\put(0,0){\line(1,0){4}}\qbezier(0,0)(0,0)(-0.2,4.4)\qbezier(4,0)(4,0)(3.8,3.5)\put(-0.2,3.5){\line(1,0){4}}\qbezier(0,0)(0,0)(3.8,3.5)\put(2.3,0.01){\vector(1,0){0}}\put(2,3.5){\vector(1,0){0}}\multiput(-0.07,2)(-0.02,0.3){2}{\vector(0,1){0}}\multiput(3.89,2)(0,0.3){2}{\vector(0,1){0}}\qbezier(0.8,0)(1,0.35)(0.5,0.5)\qbezier(3.3,3)(3.5,2.7)(3.8,2.8)\qbezier(-0.2,3.9)(0.3,3.9)(0.4,3.5)\put(1,0.3){ \bf 47^{\circ} }\put(3.1,2.3){ \bf 35^{\circ} }\put(0.3,4){ \bf 82^{\circ} }\end{picture}$

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$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\put(0,0){\line(-1,1){3}}\put(0,0){\line(1,1){3}}\put(-3,3){\line(1,1){3}}\put(3,3){\line(-1,1){3}}\put(-3,3){\vector(-1,1){1}}\put(3,3){\vector(1,-1){1}}\qbezier(-0.4,0.4)(0,0.7)(0.4,0.4)\qbezier(-0.4,5.6)(0,5.3)(0.4,5.6)\qbezier(-2.6,3.4)(-3,3.7)(-3.4,3.3)\qbezier(2.6,2.6)(3,2.3)(3.4,2.6)\put(-0.1,5){\bf\large x}\put(-0.4,0.8){ \bf 140^{\circ} }\put(2.6,2){ \bf 130^{\circ} }\put(-3.4,3.8){ \bf 140^{\circ} }\end{picture}$

See the Answer at : https://brainly.in/question/32012901