1. Write an inequality for the range of the third side of a triangle if two sides measure 4 and 13. 2. If LM = 12 and NL = 7 of ∆LMN, write an inequalty to describe the lenght of MN. 3. Use the Hinge Theorem to compare the measures of AD and BD.

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 thatThe 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 trianglesoApplying the triangle inequality theorema)  4+13 > x 17 > xRewritex < 17 unitsb) x+4 > 13x > 13-4x > 9 unitsthereforeThe inequality for the range of the third side is equal to[tex]9 < x < 17[/tex]Part 2) we know thatThe 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 trianglesoApplying the triangle inequality theorema)  LM+NL > MN 12+7 > MN 19 > MN RewriteMN < 19 unitsb) MN+NL > LMMN+7 > 12MN > 12-7MN > 5 unitsthereforeThe inequality to describe the length of MN is [tex]5 < MN < 19[/tex]   Part 3) we know thatThe 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 sideIn this problem Triangles ADC and BCD have two congruent sidesAC≅BCDC≅CD ---> is the same sideThe angle between AC and CD is 70 degreesThe angle between BC and CD is 68 degreesCompare70° > 68°thereforeAD is longer than BD