Answer :

Given: ΔABC, exterior angles ABD and ACE


To prove: the sum of the measurement of these two exterior angles is more than 2 right angles, i.e., ABD + ACE > 2(90°)


The figure for the given question is as shown below,



We know in a triangle the measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.


So in ΔABC ABD and ACE are exterior angles, so


ABD = BAC + ACB……..(i)


And,


ACE = ABC + BAC……..(ii)


Adding equation (i) an equation (ii), we get


ABD + ACE = BAC + ACB + ABC + BAC …….(iii)


We also know in a triangle the sum of all three interior angles is equal to 180°.


So in this case,


BAC + ACB + ABC = 180°


Substituting the above equation in equation (iii), we get


ABD + ACE = (BAC + ACB + ABC) + BAC


ABD + ACE = (180°) + BAC


ABD + ACE > 180°


ABD + ACE > 2(90°)


Hence the sum of the measurement of these two exterior angles is more than 2 right angles.


Hence proved


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