15. The Sketch indicates that `bar(AD)` ` ` and `bar(BC)` ` ` are parallel. Consequently, vertical angles`/_ADE~=/_BCE=40` ` `degrees. `/_ABC` is found by using the property of triangles, that the sum of the measures of the interior angles of a triangle is 180 degrees. We already have interior angles of 40 and 79, so the remaining angle should be 180-(40+79)=180-119=61 degrees. And by the same property of vertical angles m`/_EAD` =61 degrees. The two remaining angles can be found two ways: Method 1, observe that `/_DAB` is supplementary to `/_EAD` and `/_CDE` is supplementary to `/_ADE` so the sums of each pair should be 180 degrees. 180-61=119, and 180-40=140 respectively. You could also use the fact that these angles are exterior angles of a triangle, and the exterior angle of a triangle is the sum of the measures of the opposite interior angles.
16. From the property of exterior angles i.e. the measure of the exterior angle of a triangle is the sum of the measures of the opposite two angles, we find that `4x-83=2x+(x-35)` Combining like terms: `4x-83=3x-35` Isolating the variable `x=83-35` x=48
19. Angle DBF is opposite angle ABC which measures 114. Since opposite angles are congruent, Angle DBF is also 114 degrees.
20. The corresponding sides are AB:WV, AC:VX, and CB:XW. The corresponding angles, B:W, A:V, and C:X
22. According to Side-angle-side theory, `DeltaDEF~=DeltaHIJ` Then use pythagorean theorem to find IH which corresponds to EF and should be the same length. `73^2-48^2=IH^2` IH=55, so EF is also 55
18. Draw two lines that cross the provided line forming congruent vertical angles.
Since the two triangles have the same angle measures, their side lengths must be proportional to each other. In this case, the side lengths are the same for each corresponding side. Side EF is in between 73 side length and the right angle. This corresponds to side HI which is also in between the right angle and the 73 side. So, if we determine side length HI, we can determine the length of EF
(Using the Pythagorean Theorem)
HI = sqrt (73^2 - 48^2 ) = 55
Therefore, EF = 55.
What about Number 22