If m∠ABE=7x+4 and m∠EBF=5x+10. Find the m∠ABF.
Understand the Problem
The question is asking to find the measure of angle ABF given the measures of angles ABE and EBF in terms of x. It involves solving for x and then using it to determine the required angle's measure.
Answer
The measure of angle \( ABF \) is \( 180^\circ \).
Answer for screen readers
The measure of angle ( ABF ) is ( 180^\circ ).
Steps to Solve
- Identify the relationship between angles
Since angles ( ABE ) and ( EBF ) are adjacent and form a straight angle ( ABF ) at point ( B ), we can express this relationship as: $$ m∠ABF = m∠ABE + m∠EBF $$
- Set up the equation
Substituting the given expressions for the angles into the equation, we have: $$ m∠ABF = (7x + 4) + (5x + 10) $$
- Combine like terms
Combine the terms on the right side of the equation: $$ m∠ABF = 7x + 5x + 4 + 10 $$ $$ m∠ABF = 12x + 14 $$
- Use the straight angle property
Since ( m∠ABF ) is equal to ( 180^\circ ): $$ 12x + 14 = 180 $$
- Solve for ( x )
Now, isolate ( x ): $$ 12x = 180 - 14 $$ $$ 12x = 166 $$ $$ x = \frac{166}{12} $$ $$ x = \frac{83}{6} $$
- Find ( m∠ABF )
Now substitute ( x ) back into the expression for ( m∠ABF ): $$ m∠ABF = 12 \left( \frac{83}{6} \right) + 14 $$ $$ = 166 + 14 $$ $$ = 180 $$
The measure of angle ( ABF ) is ( 180^\circ ).
More Information
The angles ( ABE ) and ( EBF ) are supplementary, meaning they add up to ( 180^\circ ). This relationship is fundamental in solving the problem.
Tips
- Incorrectly summing angles: Sometimes students mistakenly subtract instead of adding the angles. Always ensure to check the relationship (adjacent, complementary, or supplementary) between the angles.
- Miscalculating ( x ): Be careful with arithmetic when isolating ( x ). Double-check each step to avoid errors.
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