Ex.6.1 Q6 Lines and Angles Solution - NCERT Maths Class 9

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Question

It is given that \(\angle XYZ = 64^ {\circ}\) and \(XY\) is produced to point \(P.\) Draw a figure from the given information. If ray \(YQ\) bisects \(\angle ZYP,\) find \(\angle XYQ\) and reflex \(\angle QYP.\)

 

 Video Solution
Lines And Angles
Ex 6.1 | Question 6

Text Solution

What is known?

\(\angle XYZ = 64^ {\circ} \) and Ray \(YQ\) bisects \(\angle PYZ.\)

What is unknown?

\(\angle XYQ = ?\) and Reflex \(\angle QYP = ?\)

Reasoning:

When a ray intersects a line sum of adjacent angles formed is \(180^ {\circ}.\)

Steps:

With the given information in the question, we can come up with this diagram.

Ray \(YQ\) bisects \(\angle ZYP\) and let \(\angle ZYQ = \angle QYP = a.\)

We can see from figure that \(PX\) a line and \(YZ\) is a ray intersecting at point \(Y\) and the sum of adjacent angles so formed is \(180^ {\circ}.\)

Hence \( \angle ZYP + \angle ZYX = 180 ^ { \circ }\)

\(\begin{align}\angle ZYQ + \angle QYP + &\angle ZYX  = 180 ^ { \circ } \\ a + a + 64 ^ {\circ } & = 180 ^ { \circ } \\ 2 a + 64 ^ {\circ } & = 180 ^ { \circ } \\ 2 a & = 180 ^ { \circ }-64 ^ {\circ } \\&= 116 ^ { \circ } \\ a & = \frac { 116 } { 2 } \\ & = 58 ^ { \circ } \end{align}\)

Then \(\angle XYQ = \angle XYZ + \angle ZYQ\) 

\[\begin{align}\angle XYQ  &= a + 64 ^ {\circ } \\  &= 58 ^ { \circ } + 64 ^ {\circ} \\  &= 122 ^ { 0 } \end{align}\]

As,

\(\begin{align}\angle QYP  &= a, \text {Reflex}\angle QYP \\&= (360 ^ { \circ } - a) \end{align}\)

\[\begin{align}&= (360 ^ { \circ } - 58 ^ { \circ }) \\ &= 302 ^ { \circ } \end{align}\]

Reflex,\(\angle QYP = 302 ^ { \circ }\) 

 Video Solution
Lines And Angles
Ex 6.1 | Question 6
  
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