Answer:
Step-by-step explanation:
1) in a rhombus, the diagonals bisect each other at the midpoint, forming 4 right angles.
MK = NK + NM
Since MK = 24 and NK = NM, then,
NK = 24/2 = 12
JL = NJ + NL
Since JL = 20 and NJ = NMlL, then,
NL = 20/2 = 10
Looking at triangle LMN, it is a right angle triangle. Applying Pythagoras theorem,
ML^2 = 10^2 + 12^2 = 100 + 144 = 244
ML = √244 = 15.62
Since all the sides of the rhombus are equal, then
MJ = ML = 15.62
Angle KNL = angle JML = 90 degrees. This is so because the diagonals are perpendicular)
Angle KJL = 90 - angle MJL
Angle KJL = 90 - 50 = 40 degrees.
Since the opposite angles of the rhombus are equal,
Angle MLK = angle angle MJK = 50 + 40 = 90 degrees
2) since all the sides of a rhombus are equal,
5x + 16 = 9x - 32
9x - 5x = 32 + 16
4x = 48
x = 48/4 = 12
Since PQ = NR = 5x + 16, then
PQ = 5×12 + 16 = 60 + 16
PQ = 76
3) Triangle XYZ is an isosceles triangle. This means that its base angles, XZY and ZXY are equal. The sum of angles in a triangle is 180 degrees. Therefore,
Angle XZY + angle ZXY = 180 - 136
Angle XZY + angle ZXY = 44
Angle XZY = 44/2 = 22
Therefore,
10x - 8 = 22
10x = 22 + 8 = 30
x = 30/10 = 3
