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નીચેની દરેક આકૃતિમાં xની કિંમત ગણો :  (x દર્શાવેલ ખૂણાનું માપ દર્શાવે છે.)


આકૃતિ : 1

અહીં, bold m bold space bold angle bold space bold BAC bold space bold plus bold space bold m bold space bold angle bold space bold BAE bold space bold equals bold space bold 180 (રૈખિક જોડના ખૂણા)

bold therefore bold space bold m bold space bold angle bold space bold BAC bold space bold plus bold space bold 110 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold BAC bold space bold equals bold space bold 180 bold space bold minus bold space bold 110

bold therefore bold space bold m bold space bold angle bold space bold BAC bold space bold equals bold space bold 70


bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold m bold space bold angle bold space bold ACD bold space bold equals bold space bold 180 (રૈખિક જોડના ખૂણા)

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold 105 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold equals bold space bold 180 bold space bold minus bold space bold 105

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold equals bold space bold 75


bold increment bold space bold ABCમાં,


bold m bold space bold angle bold space bold BAC bold space bold plus bold space bold m bold space bold angle bold space bold ABC bold space bold plus bold space bold m bold space bold angle bold angle bold space bold ACB bold space bold equals bold space bold 180

bold therefore bold space bold 70 bold space bold plus bold space bold x bold space bold plus bold space bold 75 bold space bold equals bold space bold 180

bold therefore bold space bold x bold space bold equals bold space bold 180 bold space bold minus bold space bold 70 bold space bold minus bold space bold 75

bold therefore bold space bold x bold space bold equals bold space bold 35



આકૃતિ : 2

bold m bold space bold angle bold space bold ABE bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ACB

bold therefore bold space bold 105 bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ACB bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. open parentheses bold 1 close parentheses

bold m bold space bold angle bold space bold ACD bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ABC

bold therefore bold space bold 100 bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold angle bold space bold ABC bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. open parentheses bold 2 close parentheses


પરિણામ (1) અને (2)નો સરવાળો લેતાં,

bold 105 bold space bold plus bold space bold 100 bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ABC

bold therefore bold space bold 205 bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space open parentheses bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold ABC close parentheses

bold therefore bold space bold 205 bold space bold equals bold space bold m bold space bold angle bold space bold A bold space bold plus bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold A bold space bold equals bold space bold 205 bold space bold minus bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold A bold space bold equals bold space bold 25

bold therefore bold space bold x bold space bold equals bold space bold 25


bold increment bold space bold ABCમાં, bold m bold space bold angle bold space bold A bold space bold minus bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 70 bold space bold અન ે bold space bold m bold space bold angle bold space bold B bold space bold minus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 40 હોય, તો bold increment bold space bold ABCના તમામ ખૂણાઓનાં માપ શોધો.


bold ધ ા ર ો bold space bold ક ે bold comma bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold x

bold m bold space bold angle bold space bold A bold space bold minus bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 70

bold therefore bold space bold m bold space bold angle bold space bold A bold space bold minus bold space bold x bold space bold equals bold space bold 70

bold therefore bold space bold m bold space bold angle bold space bold A bold space bold equals bold space bold x bold space bold plus bold space bold 70

bold m bold space bold angle bold space bold B bold space bold minus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 40

bold therefore bold space bold x bold space bold minus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 40

bold therefore bold space bold x bold space bold minus bold space bold 40 bold space bold equals bold space bold m bold space bold angle bold space bold C

bold therefore bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold x bold space bold minus bold space bold 40

bold increment bold space bold ABCમ ાં bold comma bold space

bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold B bold space bold plus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 180

bold therefore bold space open parentheses bold x bold space bold plus bold space bold 70 close parentheses bold space bold plus bold space bold x bold space bold plus bold space open parentheses bold x bold space bold minus bold space bold 40 close parentheses bold space bold equals bold space bold 180

bold therefore bold space bold 3 bold x bold space bold plus bold space bold 30 bold space bold equals bold space bold 180

bold therefore bold space bold 3 bold x bold space bold equals bold space bold 180 bold space bold minus bold space bold 30

bold therefore bold space bold 3 bold x bold space bold equals bold space bold 150

bold therefore bold space bold x bold space bold equals bold space bold 50

bold આમ bold comma bold space bold increment bold space bold italic A bold italic B bold italic C bold space bold મ ાં bold comma

bold m bold space bold angle bold space bold A bold space bold equals bold space bold x bold space bold plus bold space bold 70 bold space bold equals bold space bold 50 bold space bold plus bold space bold 70 bold space bold equals bold space bold 120 bold comma bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold x bold space bold equals bold space bold 50 bold space bold અન ે bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold c bold space bold minus bold space bold 40 bold space bold equals bold space bold 50 bold space bold minus bold space bold 40 bold space bold equals bold space bold 10

bold increment bold space bold ABCમ ાં bold space bold BE with bold bar on top bold space bold perpendicular bold space bold AC with bold bar on top bold space bold comma bold space bold D bold space bold element of bold space bold BC with bold bar on top  કે જેથી B – D – C. જો bold m bold space bold angle bold space bold EBC bold space bold equals bold space bold 40 bold comma bold space bold m bold space bold angle bold space bold DAC bold space bold equals bold space bold 30 bold comma bold space bold ત ો bold space bold m bold space bold angle bold space bold BCE શોધો તથા bold m bold space bold angle bold space bold ADC શોધો.

bold increment bold space bold ABC bold space bold મ ાં bold space bold BE with bold bar on top bold space bold perpendicular bold space bold AC with bold bar on top

bold therefore bold space bold m bold space bold angle bold space bold BEC bold space bold equals bold space bold 90

bold હવ ે bold comma bold space bold increment bold space bold BECમ ાં bold comma bold space

bold therefore bold space bold 90 bold space bold plus bold space bold 40 bold space bold plus bold space bold m bold space bold angle bold space bold BCE bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold BCE bold space bold equals bold space bold 180 bold space bold minus bold space bold 90 bold space bold minus bold space bold 40

bold therefore bold space bold m bold space bold angle bold space bold BCE bold space bold equals bold space bold 50


bold increment bold space bold ADCમ ાં bold comma

bold m bold space bold angle bold space bold DAC bold space bold plus bold space bold m bold space bold angle bold space bold ADC bold space bold plus bold space bold m bold space bold angle bold space bold DCA bold space bold equals bold space bold 180

bold therefore bold space bold 30 bold space bold plus bold space bold m bold space bold angle bold space bold ADC bold space bold plus bold space bold 50 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold ADC bold space bold equals bold space bold 180 bold space bold minus bold space bold 30 bold space bold minus bold space bold 50

bold therefore bold space bold m bold space bold angle bold space bold ADC bold space bold equals bold space bold 100

bold increment bold space bold ABC ના ખૂણાઓનાં માપ 2 : 3 : 5ના પ્રમાણમાં હોય, તો તમામ ખૂણાઓનાં માપ શોધો.

bold increment bold space bold ABCમ ાં bold space bold m bold space bold angle bold space bold A bold space bold colon bold space bold m bold space bold angle bold angle bold space bold B bold space bold colon bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 2 bold space bold colon bold space bold 3 bold space bold colon bold space bold 5 bold space bold છ ે bold.


આથી જો bold m bold space bold angle bold space bold A bold space bold equals bold space bold 2 bold k bold comma bold space bold ત ો bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 3 bold k bold space bold અન ે bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 5 bold k bold.

bold increment bold space bold ABC માં,

bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold B bold space bold plus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 180

bold therefore bold space bold 2 bold k bold space bold plus bold space bold 3 bold k bold space bold plus bold space bold 5 bold k bold space bold equals bold space bold 180

bold therefore bold space bold 10 bold k bold space bold equals bold space bold 180

bold therefore bold space bold k bold space bold equals bold space bold 180 over bold 10

bold therefore bold space bold k bold space bold equals bold space bold 18

bold therefore bold space bold m bold space bold angle bold space bold A bold space bold equals bold space bold 2 bold k bold space bold equals bold space bold 2 bold space bold cross times bold space bold 18 bold space bold equals bold space bold 36

bold space bold space bold space bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 3 bold k bold space bold equals bold space bold 3 bold space bold cross times bold space bold 18 bold space bold equals bold space bold 54

bold space bold space bold space bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 5 bold k bold space bold equals bold space bold 5 bold space bold cross times bold space bold 18 bold space bold equals bold space bold 90



આમ, bold increment bold space bold ABCના ખૂણાઓનાં માપ 36, 54 અને 90 છે.


જો bold increment bold space bold ABC ના બહિષ્કોણ bold angle bold space bold ACD નું માપ 120 છે અને તેના બે અંતઃસંમુખકોણ પૈકી કોઈ એકનું માપ 40 છે, તો bold increment bold space bold ABC ના અન્ય ખૂણાનાં માપ શોધો.





bold increment bold space bold ABCમ ાં bold space bold angle bold space bold ACD એ બહિષ્કોણ છે.

bold m bold space bold angle bold space bold ACD bold space bold equals bold space bold 120 છે.

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold m bold space bold angle bold space bold ACD bold space bold equals bold space bold 180  (રૈખિક જોડના ખૂણા)

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold plus bold space bold 120 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold equals bold space bold 180 bold space bold minus bold space bold 120

bold therefore bold space bold m bold space bold angle bold space bold ACB bold space bold equals bold space bold 60



અર્થાત્ bold increment bold space bold ABCમ ાં bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 60 છે.

હવે, bold increment bold space bold ABC માં,

bold m bold space bold angle bold space bold A bold space bold plus bold space bold m bold space bold angle bold space bold B bold space bold plus bold space bold m bold space bold angle bold space bold C bold space bold equals bold space bold 180

bold therefore bold space bold 40 bold space bold plus bold space bold m bold space bold angle bold space bold B bold space bold plus bold space bold m bold space bold angle bold space bold 60 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold B bold space bold plus bold space bold 100 bold space bold equals bold space bold 180

bold therefore bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 180 bold space bold minus bold space bold 100

bold therefore bold space bold m bold space bold angle bold space bold B bold space bold equals bold space bold 80



bold therefore bold space bold increment bold space bold ABC ના બાકીના બે ખૂણાઓનાં માપ 80 અને 60 છે.