Answer:
x = 5
Explanation:
The diagram shows that the rhombus is split into two isosceles triangles, LKM and NMK.
- Isosceles triangles have two sides equal in length and the angles opposite these sides are always congruent and equal.
Thus, the three angles in triangle LKM are 110, (8x - 5), and (8x - 5).
- The Triangle Angle Sum Theorem says that the sum of the measures of the interior angles in a triangle always equals 180°.
Thus, we can solve for x by setting the sum of the measures of the three angles in triangle LKM equal to 180:
(8x - 5) + (8x - 5) + 110 = 180
(8x + 8x) + (-5 - 5 + 110) = 180
16x + 100 = 180
16x = 80
x = 5
Thus, x = 5
Optional step:
We can check that we've correctly solved for x by plugging in 5 for x in (8x - 5) twice for both angles, adding the result to 110, and seeing if we get 180 on both sides of the equation:
(8(5) - 5) + (8(5) - 5) + 110 = 180
(40 - 5) + (40 - 5) + 110 = 180
35 + 35 + 110 = 180
70 + 110 = 180
180 = 180
Thus, x = 5 is correct.