BE bisects ∠ABF. If m∠ABE = 7x - 12 and m∠EBF = 4x + 9, find the m∠ABE.
Understand the Problem
The question is asking to find the measure of angle ABE given that BE bisects angle ABF. It provides two expressions for the measures of the angles and sets them equal because the bisector divides the angle into two equal parts.
Answer
The measure of $\angle ABE$ is $37^\circ$.
Answer for screen readers
The measure of $\angle ABE$ is $37$ degrees.
Steps to Solve
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Set up the equation for the bisector Since BE bisects $\angle ABF$, it means that the measures of the two angles created by the bisector are equal. Therefore, we can set the two expressions equal to each other: $$ m\angle ABE = m\angle EBF $$ This translates to: $$ 7x - 12 = 4x + 9 $$
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Isolate the variable x To solve for $x$, we need to isolate it on one side of the equation. Start by subtracting $4x$ from both sides: $$ 7x - 4x - 12 = 9 $$ Simplify the left side: $$ 3x - 12 = 9 $$
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Solve for x Next, add 12 to both sides to isolate the term with $x$: $$ 3x = 21 $$ Now divide by 3: $$ x = 7 $$
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Substitute back to find m∠ABE Now that we have the value for $x$, we can substitute it back into the expression for $m\angle ABE$: $$ m\angle ABE = 7(7) - 12 $$ Calculating that gives us: $$ m\angle ABE = 49 - 12 = 37 $$
The measure of $\angle ABE$ is $37$ degrees.
More Information
In this problem, we utilized the property of angle bisectors, which states that the angles created are equal. Solving linear equations was essential in determining the value of the variable.
Tips
- Forgetting to set both expressions equal due to the angle bisector property.
- Miscalculating while isolating the variable or substituting it back into the angle equation. To avoid this, double-check calculations at each step.
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