prove: a) the sum of the measures of any two angles of a trianlge is less than 180. b) a triangle can have at most one right or obtuse angle c) the base angles of an isocseles triangle are acute. Solution A) Triangle ABC - draw a line extending horizontally from C, name that end point D. angle ABC < angle ACD {prop 16} angle ABC + angle ACB < angle ACD + angle ACB But angles ACD + ACB = 180 degrees Therefore angle ABC + angle ACB < 180 degrees - Similar logix can be used for any two angles of the triangle B) A triangle can\'t have more than one obtuse angle because all the angles have to add up to 180 degrees. More than one obtuse angle would be more than 180 degrees. There would not be enough room for another angle. C) Isosceles implies 2 base angles are identical Call their measures x sum of base angles is 2x sum of all three angles is 180 degrees call third angle\'s measure y 2x+y=180 2x=180-y but y>0 2x<180 x<90.

1. prove:
a) the sum of the measures of any two angles of a trianlge is less than 180.
b) a triangle can have at most one right or obtuse angle
c) the base angles of an isocseles triangle are acute.
Solution
A)
Triangle ABC
- draw a line extending horizontally from C, name that end point D.
angle ABC < angle ACD {prop 16}
angle ABC + angle ACB < angle ACD + angle ACB
But angles ACD + ACB = 180 degrees
Therefore angle ABC + angle ACB < 180 degrees
- Similar logix can be used for any two angles of the triangle
B)
A triangle can't have more than one obtuse angle because all the angles have to add up to 180
degrees. More than one obtuse angle would be more than 180 degrees. There would not be
enough room for another angle.
C)
Isosceles implies 2 base angles are identical
Call their measures x
sum of base angles is 2x
sum of all three angles is 180 degrees
call third angle's measure y
2x+y=180
2x=180-y
but y>0
2x<180

2. x<90