In the figure, find

Explanation

Strategy

We will use the property of vertical angles to write an equation that mathematically models the situation.

WHY

\(\overleftrightarrow{AE}\) and \(\overleftrightarrow{DC}\) intersect to form two pairs of vertical angles.

Solution

1. In the figure, two vertical angles have degree measures that are represented by the algebraic expressions 4x - 20^o and 3x + 15^o.  Since the angles are vertical angles, they have equal measures
   • Set the algebraic expressions equal
     • 4x - 20^o = 3x + 15^o
   • To eliminate 3x from the right side, subtract 3x from both sides
     • 4x - 20^o - 3x = 3x + 15^o - 3x - 3x
   • Combine like terms: 4x - 3x = x and 3x - 3x = 0
     • x - 20^o = 15^o
   • To isolate x, undo the subtraction of 20^o by adding 20^o to both sides
     • x = 35^o
2. To find m(∠ABC), we evaluate the expression 3x + 15^o for x = 35^o
   • Substitute 35^o for x
     • 3x + 15^o = 3(35^o) + 15^o
   • Do the multiplication
     • = 105^o + 15^o
   • Do the addition
     • = 120^o
   • Thus, m(∠ABC) = 120^o
3. ∠ABE is a straight angle.  Since the measure of a straight angle is 180^o and m(∠ABC) = 120^o, m(∠CBE) must be 180^o - 120^o, or 60^o