Answer:
Part 1) The inequality for the range of the third side is [tex]9 < x < 17[/tex]
Part 2) The inequality to describe the length of MN is [tex]5 < MN < 19[/tex]
Part 3) AD is longer than BD (see the explanation)
Step-by-step explanation:
Part 1) we know that
The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.
Let
x ----> the measure of the third side of a triangle
so
Applying the triangle inequality theorem
a) 4+13 > x
17 > x
Rewrite
x < 17 units
b) x+4 > 13
x > 13-4
x > 9 units
therefore
The inequality for the range of the third side is equal to
[tex]9 < x < 17[/tex]
Part 2) we know that
The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.
Let
x ----> the measure of the third side of a triangle
so
Applying the triangle inequality theorem
a) LM+NL > MN
12+7 > MN
19 > MN
Rewrite
MN < 19 units
b) MN+NL > LM
MN+7 > 12
MN > 12-7
MN > 5 units
therefore
The inequality to describe the length of MN is
[tex]5 < MN < 19[/tex]
Part 3) we know that
The hinge theorem states that if two triangles have two congruent sides, then the triangle with the larger angle between those sides will have a longer third side
In this problem Triangles ADC and BCD have two congruent sides
AC≅BC
DC≅CD ---> is the same side
The angle between AC and CD is 70 degrees
The angle between BC and CD is 68 degrees
Compare
70° > 68°
therefore
AD is longer than BD
