In the trapezoid below, if m∠𝐵𝐴𝐶∠𝐵𝐴𝐶 = (3x - 4)° and m∠𝐷𝐶𝐴∠𝐷𝐶𝐴 is (2x + 9)°, find x.

1. Break Down the Problem

We have a trapezoid where two angles are related to the variable $x$. We need to find the value of $x$ given the angles:

• $m∠BAC = (3x - 4)°$
• $m∠DCA = (2x + 9)°$

In a trapezoid, consecutive angles (i.e., angles on the same side of a leg) are supplementary. This means their sum will equal $180°$.

2. Relevant Concepts

Using the property of trapezoids: $m∠BAC + m∠DCA = 180°$

3. Analysis and Detail

Substituting the expressions for the angles into the equation gives us: $(3x - 4) + (2x + 9) = 180$

Now, we'll simplify and solve for $x$: $3x - 4 + 2x + 9 = 180$ Combine like terms: $5x + 5 = 180$ Next, we isolate $x$ by subtracting $5$ from both sides: $5x = 175$ Now divide both sides by $5$: $x = 35$

4. Verify and Summarize

To verify, substitute $x = 35$ back into the angle expressions: $m∠BAC = 3(35) - 4 = 105 - 4 = 101°$ $m∠DCA = 2(35) + 9 = 70 + 9 = 79°$ Now check if they are supplementary: $101° + 79° = 180°$ This confirms our solution as both angles sum to $180°$.

Final Answer

The value of $x$ is $35$.