if m<GEF = 2x + 10, m<DEF = 5x- 10, and m<GED = 70, find m<GEF and m<DEF
To solve this problem, we need to use the angle relationships within the given triangle. Let's break down the steps to find the measures of angles GEF and DEF:
Step 1: Understand the angle relationships in a triangle
In a triangle, the sum of the interior angles equals 180 degrees. So, we can use this fact to find the missing angle measures.
Step 2: Write an equation for the given angle relationship
We are given that m<GEF is equal to 2x + 10 and m<DEF is equal to 5x - 10. We'll use these expressions to write an equation.
Step 3: Set up an equation using the angle relationship
Since GEF is adjacent to both GED and DEF, we know that m<GEF + m<GED + m<DEF = 180 degrees. Substitute the given measures and known variables into this equation.
m<GEF + 70 + m<DEF = 180
Step 4: Simplify and solve the equation
Combine like terms and isolate the variable on one side. Then solve for x.
2x + 10 + 70 + 5x - 10 = 180
7x + 70 = 180
7x = 180 - 70
7x = 110
x = 110/7
x ≈ 15.71
Step 5: Find the angle measures
Substitute the value of x back into the expressions for m<GEF and m<DEF.
m<GEF = 2x + 10
m<GEF = 2(15.71) + 10
m<GEF ≈ 41.42
m<DEF = 5x - 10
m<DEF = 5(15.71) - 10
m<DEF ≈ 73.55
Therefore, m<GEF is approximately 41.42 degrees and m<DEF is approximately 73.55 degrees.