Answer:
10. JL = 4
11. m∠KML = 45°
12. see below for a full list of angle measures; x = 30
Explanation:
10. In the square, the diagonals are both the same length, and point N is their midpoint. That means ...
KM = 2·JN
4z -8 = 2(4z -10) . . . . substitute the given expressions
4z -8 = 8z -20 . . . . . .eliminate parentheses
12 = 4z . . . . . . . . . . . .add 20-4z to both sides
KM = 12 -8 = 4 . . . . . substitute for 4z in the expression for KM
JL = KM = 4
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11. All the sides and angles that look like they're the same measure are the same measure. Any angle not 90° is 45°. m∠KML = 45°
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12. For this, you make use of the properties of a rhombus. The diagonals of a rhombus are perpendicular bisectors of each other, so all of the angles where the diagonals meet are 90°.
Then within each triangle, the angles not 50° are their complement, 40°.
Angles in opposite corners are congruent to each other, and each diagonal bisects the corner angle. So ...
∠GHJ = ∠GKJ = 100°
∠GHK = ∠JHK = ∠GKH = ∠JKH = 50°
∠HGK = ∠HJK = 80°
∠HJG = ∠KJG = ∠KGJ = ∠KJG = 40°
We have already said the central angle is 90°, so ...
(5x -60)° = 90°
x -12 = 18 . . . . . . divide by 5°
x = 30 . . . . . . . . add 12