# 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...

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.

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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.

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Sum of all angles in triangle 180.

so 93+43+angle(dba)=180

angle(DBA)=44

angle(BDC)+angle (DCB)+angle(DBC)=180

x+19+3x+5+2x+6=180

6x+30=180

6x=150

x=25

angle(BDC)= x+19=25+19=44

But it is alternate angle to angle DBA

therefore CD is parallel to BA