AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Solve for x and Angle Measures with Perpendicular Lines and Vertical Angles
Mathematics
Grade 7 (Junior High School)
Question Content
In the given diagram, line VF is perpendicular to line AR, forming a right angle at point F. We know that ∠VFT = 36°, ∠BFG = 2x + 9, and ∠GFR = 4x - 3. Solve for x, ∠BFG, ∠GFR, and confirm ∠TFR.
Correct Answer
x = 22, ∠BFG = 53°, ∠GFR = 85°, ∠TFR = 54°
Detailed Solution Steps
1
Step 1: Calculate ∠TFR. Since VF ⊥ AR, ∠VFR = 90°. We know ∠VFT = 36°, so ∠TFR = ∠VFR - ∠VFT = 90° - 36° = 54°.
2
Step 2: Set up the equation for x. The angles on the straight line AR at point F add up to 180°, so ∠AFB + ∠BFG + ∠GFR = 180°. Also, ∠AFB and ∠TFR are vertical angles, so ∠AFB = ∠TFR = 54°. Substitute the known values: 54° + (2x + 9) + (4x - 3) = 180°.
3
Step 3: Simplify and solve for x. Combine like terms: 54 + 9 - 3 + 2x + 4x = 180 → 60 + 6x = 180. Subtract 60 from both sides: 6x = 120. Divide by 6: x = 22.
4
Step 4: Calculate ∠BFG. Substitute x = 22 into 2x + 9: 2(22) + 9 = 44 + 9 = 53°.
5
Step 5: Calculate ∠GFR. Substitute x = 22 into 4x - 3: 4(22) - 3 = 88 - 3 = 85°.
6
Step 6: Verify. Check that 54° + 53° + 85° = 180°, which confirms the solution is correct.
Knowledge Points Involved
1
Vertical Angles Theorem
Vertical angles are the opposite angles formed when two lines intersect. This theorem states that vertical angles are congruent (equal in measure). In this problem, ∠AFB and ∠TFR are vertical angles, so they have the same degree measure.
2
Perpendicular Lines and Right Angles
Two lines are perpendicular if they intersect to form a right angle (90°). Here, VF ⊥ AR, so ∠VFA and ∠VFR are both 90°, which is used to calculate ∠TFR.
3
Straight Angle Sum Property
A straight angle measures 180°. All angles that lie on a straight line (forming a linear pair or a straight angle sum) add up to 180°. This is used to set up the equation to solve for x, as ∠AFB + ∠BFG + ∠GFR form a straight line.
4
Solving Linear Equations with One Variable
This involves simplifying algebraic expressions, combining like terms, and using inverse operations to isolate the variable. It is used here to solve the equation 60 + 6x = 180 to find the value of x.
Loading solution...
