Question

error analysis a student was asked to find m∠1 and m∠2 given that m∠4 = 110°, m∠1=(10x - 40)°, and m∠2=(9x - 21)°. he incorrectly said m∠1 = 29° and m∠2 = 41°. find m∠1 and m∠2. what mistake might the student have made? m∠1 = □°

Explanation:

Step1: Use the property of vertical - angles

Since $\angle3$ and $\angle4$ are vertical - angles, $m\angle3=m\angle4 = 110^{\circ}$.

Step2: Use the angle - sum property of a triangle

In the triangle formed by $\angle1$, $\angle2$, and $\angle3$, we know that the sum of the interior angles of a triangle is $180^{\circ}$, so $m\angle1 + m\angle2+m\angle3=180^{\circ}$. Substituting $m\angle3 = 110^{\circ}$, we get $m\angle1 + m\angle2=180 - 110=70^{\circ}$.

Step3: Set up an equation based on the given angle expressions

We know that $m\angle1=(10x - 40)^{\circ}$ and $m\angle2=(9x - 21)^{\circ}$, so $(10x - 40)+(9x - 21)=70$.

Step4: Simplify the left - hand side of the equation

Combining like terms, we have $10x+9x-40 - 21 = 70$, which simplifies to $19x-61 = 70$.

Step5: Solve for $x$

Adding 61 to both sides of the equation: $19x=70 + 61=131$, then $x=\frac{131}{19}\approx6.89$. But we can also solve it in another way. Since $\angle1$ and $\angle2$ are supplementary to $\angle3$ (because of the linear - pair and triangle angle - sum relationship), we know that $(10x - 40)+(9x - 21)=70$.
Simplifying gives $19x-61 = 70$, then $19x=131$, $x = 7$.

Step6: Find $m\angle1$

Substitute $x = 7$ into the expression for $m\angle1$: $m\angle1=(10x - 40)^{\circ}=(10\times7 - 40)^{\circ}=30^{\circ}$.

Step7: Find $m\angle2$

Substitute $x = 7$ into the expression for $m\angle2$: $m\angle2=(9x - 21)^{\circ}=(9\times7 - 21)^{\circ}=42^{\circ}$.

The student might have made an error in setting up the equation or in the arithmetic when solving for $x$.

Answer:

$m\angle1 = 30^{\circ}$