+91 1) In triangle ABC, [ADE] = 4, [ABE] = 14, and [CDE] = 6. Compute [BCE].
2) In the diagram, WY = 12, XZ = 10, [AWX] = 15, and [AYZ] = 8. Find [AXY]
\(\text{In the diagram below, we have } \overline{AB}\parallel\overline{CD}, EF = FG, \angle AEG = x^\circ, \text{and } \angle BEF = 102^\circ + x^\circ. \text{Find the value of x.}\)
In triangle ABC, AB = AC. Find
+1622 1)
ADE and CDE share the same height, but different bases, given the ratio of areas being 4 : 6 (ADE : CDE), we can conclude that the ratio of AD:DC is 2:3
Also, the area of triangle ABD is 14 + 4 = 18 units squared.
The triangles ABD and BCD share the same height, so the ratio of their areas is the ratio of AD:DC, which is 2:3, then we can conclude that triangle BCD has area of 27 units squared. Subtracting 6, the area of triangle BCE is 21 units squared.
+1622 2)
Area of triangle AXY now has area "x".
Since all the triangles in the image share the same height (the perpendicular from apex A to WZ), then the ratio of the triangles' areas are the ratio of their bases. WY/XZ = 12/10, so the areas of triangles AWY/AXZ = WY/XZ = 12/10.
Substituting in, we have 15 + x / 8 + x = 12/10
150 + 10x = 96 + 12x
54 = 2x
x = 27. Triangle AXY has area of 27 units squared.
+1622 3) In the diagram below, we have.....
Since AB || CD, by alternate interior angles, angle AEG = angle EGF = x*.
Also by alternate interior angles, angle BEF = angle EFG = 102 + x*.
Also EF = FG so the triangle within the two parallel lines is isosceles, by Isosceles Triangle Theorem, angle GEF = angle EGF = x.
The interior angle sum of a triangle is 180 degrees, so 102 + x + x + x = 180. 3x = 78, x = 26.
+1622 4) You did not fully fill out the question --- please re-post/edit the bottom question since it got cut off after the "Find".
+1622 4) If angle DCB is x, then angle BDC is also x.
Angle DBC = 180 - 2x
Angle DBE = x - (180 - 2x) = 3x - 180 degrees
Angle EDB = 3x - 180
Angle BED = 180 - 2(3x - 180) = 180 - 6x + 360 = 540 - 6x degrees
Angle AED = 180 - (540 - 6x) = 6x - 360 degrees
Angle A = 6x - 360 degrees.
Angle B + Angle C + Angle A = 180
x + x + 6x - 360 = 180
8x = 540
Angle BAC = 67.5 degrees
