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.
2 Answers | Add Yours
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
We’ve answered 330,629 questions. We can answer yours, too.Ask a question