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ABCD is a rhombus with diagonals that intersect at
F. Find the value of '
x ' if
m/_ABD=(2x+3)^(@) and
m/_DBC=(4x-1)^(@)

$A B C D$ is a rhombus with diagonals that intersect at $F$ . Find the value of ' $x$ ' if $m \angle A B D=(2 x+3)^{\circ}$ and $m \angle D B C=(4 x-1)^{\circ}$

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Conjugate root theorems

Full solution

Q.

A B C D

is a rhombus with diagonals that intersect at

F

. Find the value of '

x

' if

m \angle A B D=(2 x+3)^{\circ}

and

m \angle D B C=(4 x-1)^{\circ}

Rhombus Properties: In a rhombus, the diagonals bisect the angles at the vertices. This means that the angles $m/\angle ABD$ and $m/\angle DBC$ are actually two parts of the same angle, which is bisected by one of the diagonals. Therefore, the sum of $m/\angle ABD$ and $m/\angle DBC$ is equal to the measure of the whole angle at vertex $B$ .
Equation Setup: Let's set up the equation using the given angle measures. We know that $m/_{ABD} = (2x + 3)$ degrees and $m/_{DBC} = (4x - 1)$ degrees. Since these two angles add up to the measure of angle ABC, we have: $(2x + 3) + (4x - 1) = 180$ degrees (because the sum of the measures of the angles in a rhombus is $180$ degrees).
Combine Like Terms: Now, let's solve for $x$ . Combine like terms: $\newline$ $2x + 4x + 3 - 1 = 180$ $\newline$ $6x + 2 = 180$
Subtract $2$ : Subtract $2$ from both sides of the equation: $\newline$ $6x + 2 - 2 = 180 - 2$ $\newline$ $6x = 178$
Divide by $6$ : Divide both sides by $6$ to solve for x: $\newline$ $\frac{6x}{6} = \frac{178}{6}$ $\newline$ $x = 29.6666667$
Recalculate: However, since we are dealing with angle measures, we expect $x$ to be a whole number. Let's check our calculations again. It seems that we have made a mistake in the division step. Let's correct it. $\newline$ $6x = 178$ $\newline$ $x = \frac{178}{6}$ $\newline$ $x = 29.6666667$ $\newline$ This is not a whole number, which is unexpected for this kind of problem. We need to recheck our calculations.

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