Find the values of x and y.

Question

17-03-2022
Mathematics

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Find the values of x and y.

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Starrysoul100 Starrysoul100 · Answer 1 · 2022-03-17T09:27:48+01:00

[tex]\bold{\huge{\underline{ Solution }}}[/tex]

Let consider the given triangle be ABC

According to the question,

AD is the median of the triangle ABC and CE also divides the triangle into two parts. At F both the lines are intersecting.

Here We have ,

[tex]\sf{\angle{ EAF = 25{\degree}}}[/tex]
[tex]\sf{\angle{ AEF = x{\degree}}}[/tex]
[tex]\sf{\angle{ EBD = y {\degree}}}[/tex]
[tex]\sf{\angle{ FDC = 90{\degree}}}[/tex]
[tex]\sf{\angle{ DCF = 20{\degree}}}[/tex]

In triangle FDC, By using Angle sum property

[tex]\sf{ {\angle}FDC + {\angle}DCF + {\angle}CFD = 180{\degree}}[/tex]

Subsitute the required values,

[tex]\sf{ 90{\degree} + 20{\degree} + {\angle}CFD = 180{\degree}}[/tex]

[tex]\sf{ 110{\degree} + {\angle}CFD = 180{\degree}}[/tex]

[tex]\sf{ {\angle}CFD = 180{\degree} - 110{\degree}}[/tex]

[tex]\sf{ {\angle}CFD = 70{\degree}}[/tex]

Here.

[tex]\sf{ {\angle}CFD = {\angle}EFA = 70{\degree}}[/tex]
The above angles are vertically opposite angles and vertically opposite angles are equal.

Now, Again by using Angle sum property but in triangle AEF

[tex]\sf{ {\angle}AEF + {\angle}EFA + {\angle}FAE = 180{\degree}}[/tex]

Subsitute the required values,

[tex]\sf{ x{\degree} + 70{\degree} + 25{\degree} = 180{\degree}}[/tex]

[tex]\sf{ x{\degree} + 95{\degree}= 180{\degree}}[/tex]

[tex]\sf{ x{\degree} = 180{\degree} - 95{\degree}}[/tex]

[tex]\bold{ x = 85{\degree}}[/tex]

Thus , The value of x is 85°

Here, In the triangle ABC , CE divides the triangle into two parts

So ,

[tex]\sf{ {\angle}AEF + {\angle}FEB = 180{\degree}}[/tex]

[tex]\sf{ 85{\degree} + {\angle}FEB = 180{\degree}}[/tex]

[tex]\sf{ {\angle}FEB = 180{\degree} - 85{\degree}}[/tex]

[tex]\sf{ {\angle}FEB = 95{\degree}}[/tex]

Similarly ,

[tex]\sf{ {\angle}AFE + {\angle}DFE = 180{\degree}}[/tex]

Straight line angles

[tex]\sf{ 70{\degree} + {\angle}DFE = 180{\degree}}[/tex]

[tex]\sf{ {\angle}DFE = 180{\degree} - 70{\degree}}[/tex]

[tex]\sf{ {\angle}DFE = 110 {\degree}}[/tex]

In triangle ABC, EBDF is forming quadrilateral

We know that,

Sum of all the angles of quadrilateral is equal to 360°

That is ,

[tex]\sf{ {\angle}FEB + {\angle}EBD + {\angle}BDF + {\angle} DFE = 360{\degree}}[/tex]

Subsitute the required values,

[tex]\sf{ 95{\degree} + y{\degree} + 90{\degree} + + 110{\degree} = 360{\degree}}[/tex]

[tex]\sf{ y{\degree} + 295{\degree}= 180{\degree}}[/tex]

[tex]\sf{ y {\degree} = 360{\degree} - 295{\degree}}[/tex]

[tex]\bold{ y = 65{\degree}}[/tex]

Hence, The value of x and y is 70° and 65° .