we know that
1) The sum of the internal angles in a triangle is equal to [tex]180\°[/tex]
so
[tex]x+y+z=180\°[/tex] ---------> equation [tex]1[/tex]
2) The sum of the angle z and angle W is equal to [tex]180\°[/tex] by supplementary angles
so
[tex]z+w=180\°[/tex]
[tex]w=180\°-z[/tex] ---------> equation [tex]2[/tex]
Statements
case A. [tex]w > y[/tex]
This statement is true
Because
[tex]w> 90\°[/tex]
[tex]y < 90\°[/tex]
so
[tex]w > y[/tex]
case B. [tex]w > x[/tex]
This statement is true
Because
[tex]w> 90\°[/tex]
[tex]x < 90\°[/tex]
so
[tex]w > x[/tex]
case C. [tex]y+z=w[/tex] ---------> equation [tex]3[/tex]
This statement is False
Because
we know that
[tex]w=180\°-z[/tex]
substitute the value of w in the equation [tex]3[/tex]
[tex]y+z=180\°-z[/tex]
[tex]y+z+z=180\°[/tex] --------> equation [tex]4[/tex]
equate equation [tex]1[/tex] and equation [tex]4[/tex]
[tex]x+y+z=y+z+z[/tex]
[tex]x=z[/tex] ---------> not necessarily must be true
case D. [tex]x+y=z[/tex] --------> equation [tex]5[/tex]
This statement is False
Because
remember equation [tex]1[/tex]
[tex]x+y+z=180\°[/tex]
[tex]x+y=180\°-z[/tex] --------> equation [tex]6[/tex]
equate equation [tex]6[/tex] and equation [tex]5[/tex]
[tex]z=180\°-z[/tex]
[tex]2z=180\°[/tex]
[tex]z=90\°[/tex] ---------> is not true
case E. [tex]x+y=w[/tex] ---------> equation [tex]7[/tex]
This statement is True
Because
equation [tex]1[/tex]
[tex]x+y+z=180\°[/tex]
[tex]x+y=180\°-z[/tex] --------> equation [tex]8[/tex]
remember
[tex]w=180\°-z[/tex]
substitute the value of w in the equation [tex]8[/tex]
[tex]x+y=w[/tex]
case F. [tex]z > x[/tex]
This statement is False
Because
[tex]z < 90\°[/tex]
[tex]x < 90\°[/tex]
so
[tex]z > x[/tex] ---------> not necessarily must be true
