In quadrilateral ABCD with diagonal BD, measure of angle A=93, measure of angle ADB=43, measure of angle C=3x+5, measure of angle BDC=x+19, and measure of angle DBC=2x+6. Determine if AB is parallel to DC. Explain your reasoning.
The diagonal BD separates the quadrilateral into triangles: `Delta` ABD and `Delta` BCD. The sum of the angles of a triangle is equal to 180 degrees.
Two of the angles of `Delta` ABD are 93 degrees and 43 degrees. Subtract from 180 degrees to find the measure of the third angle: 180 - 93 - 43 = 44. The measure of `/_` ABD = 44 degrees.
The angles of `Delta` BCD are (x + 19), (2x + 6), and (3x + 5). Their sum is 180 degrees. Write this as an equation and solve for x.
(x + 19) + (2x + 6) + (3x + 5) = 180
6x + 30 = 180
6x = 150
x = 25
Substitute 25 in for x to find the measures of the angles of `Delta` BCD.
`/_` BDC = x + 19 = 25 + 19 = 44 degrees
`/_` DBC = 2x + 6 = 2 * 25 + 6 = 56 degrees
`/_` BCD = 3x + 5 = 3 * 25 + 5 = 80 degrees
The diagonal BD acts as a transversal, causing `/_` ABD and `` BDC to be alternate interior angles.
`/_` ABD = 44 degrees
`/_` BDC = 44 degrees
Therefore `/_` ABD = `/_` BDC.
When alternate interior angles are congruent, the opposite sides are parallel. Therefore, side AB is parallel to side CD.
However, the shape is not a parallelogram because in a parallelogram, opposite angles are congruent.
`/_` A = 93 degrees and `/_` C = 80 degrees. `/_` A `!=` `/_` C
`` B = 100 degrees and `/_` D = 87 degrees. `/_` B `` `` D
Quadrilateral ABCD is not a parallelogram (2 pairs of parallel sides). It is a trapezoid (1 pair of parallel sides).
Answer: AB is parallel to DC.
Sum of all angles in triangle 180.
But it is alternate angle to angle DBA
therefore CD is parallel to BA