Vertical Angles and Solving Equations
Vertical angles and solving equations is a Grade 7 geometry skill in Big Ideas Math Advanced 2, Chapter 12: Constructions and Scale Drawings. Since vertical angles are always congruent, you can set their algebraic expressions equal to each other and solve for the variable. For example, if one angle is (3x plus 12) degrees and its vertical angle is 72 degrees, solve 3x plus 12 equals 72 to get x equals 20.
Key Concepts
Property Vertical Angles are the opposite angles formed when two straight lines intersect to form an "X". Key Rule: Vertical angles are ALWAYS congruent (they have the exact same measure). If you know one angle, you can set up an algebraic equation to find its opposite.
Examples Basic: Two lines intersect. If the top angle is 45°, the bottom angle (its vertical opposite) is also 45°. Algebraic 1: One vertical angle is 72° and its opposite is $(3x + 12)$°. Because they are equal, you set up the equation: $$3x + 12 = 72$$ Solving this gives $3x = 60$, so $x = 20$. Algebraic 2: Two vertical angles measure $(5x 6)$° and $(3x + 10)$°. Set them equal: $$5x 6 = 3x + 10$$ Solving this gives $2x = 16$, so $x = 8$. Each angle measures 34°.
Explanation Whenever two straight lines cross, they create four angles. The ones looking directly across at each other are vertical angles, and they are perfect twins! Because we know these twins always have the exact same degree measure, we can use an equals sign to connect their algebraic expressions and solve for the missing variable.
Common Questions
How do you use vertical angles to solve for a variable?
Since vertical angles are always congruent, set their algebraic expressions equal to each other and solve the resulting equation. For example, if angles are (5x minus 6) and (3x plus 10), solve 5x minus 6 equals 3x plus 10.
What property of vertical angles lets you create equations?
Vertical angles are always congruent — they have equal measures. This equality lets you write an equation by setting two vertical angle expressions equal to each other.
What are vertical angles?
Vertical angles are the opposite angles formed when two straight lines intersect, creating an X shape. The angles directly across from each other at the intersection are vertical and always have the same measure.
What textbook covers vertical angles and equations in Grade 7?
Big Ideas Math Advanced 2, Chapter 12: Constructions and Scale Drawings covers using the vertical angles property to set up and solve algebraic equations.