In the diagram below of triangle L, M, NLMN, OO is the midpoint of start overline, L, N, end overline
LN
and PP is the midpoint of start overline, M, N, end overline
MN
. If mangle, N, M, L, equals, minus, 6, x, plus, 78∠NML=−6x+78, and mangle, N, P, O, equals, 8, x, minus, 34∠NPO=8x−34, what is the measure of angle, N, M, L∠NML?
Since OO is the midpoint of LN, it divides LN into two equal parts. Similarly, PP is the midpoint of MN, so it divides MN into two equal parts. This means that ON is equal to NO and PM is equal to PN.
Since OO is the midpoint of LN, triangle ONM is isosceles. This means that angle NMO is equal to angle NOM.
From this information, we can say:
angle NML = angle NMO + angle NMO = 2(angle NMO)
Substitute angle NMO = angle NOM:
angle NML = 2(angle NOM)
Since ON = NO and PP = PM, we can say that triangles OPM and NMO are congruent. This means that angle NPM is equal to angle NOM.
From this information, we can rewrite angle NML as:
angle NML = 2(angle NPM)
Now we can solve for angle NOM and NPM using the given equations.
angle NML = 2(8x - 34) = 16x - 68
angle NML = -6x + 78
Now equate the two expressions and solve for x:
16x - 68 = -6x + 78
22x = 146
x = 146/22
x = 73/11
Now substitute x back into the equation for angle NML:
angle NML = 16(73/11) - 68
angle NML = 105.09
Therefore, the measure of angle NML is approximately 105.09 degrees.