Question

properties of a triangle
lesson 4: interior and exterior angles

1. fill in the blanks to complete the paragraph

proof of the exterior angle theorem.

1. practice in the diagram below,

m∠efg=(30x)°,m∠e=(x² + 40)°, and
m∠d=(10x + 35)°, find the measure of ∠efg.

Explanation:

Step1: Apply the Exterior Angle Theorem

The Exterior Angle Theorem states that \(m\angle EFG=m\angle E + m\angle D\). So we set up the equation \(30x=(x^{2}+40)+(10x + 35)\).

Step2: Simplify the equation

$$
LATEXBLOCK0
$$

Step3: Factor the quadratic equation

We factor \(x^{2}-20x + 75=(x - 15)(x - 5)=0\).

Step4: Solve for \(x\)

Set each factor equal to zero:

• If \(x - 15=0\), then \(x = 15\).
• If \(x - 5=0\), then \(x = 5\).

Step5: Check for valid solutions

• When \(x = 15\):

\(m\angle EFG=30x=30\times15 = 450^{\circ}\) (not valid as an angle in a triangle - related context, since angles in a triangle - related exterior angle situation should be less than \(180^{\circ}\))

• When \(x = 5\):

\(m\angle E=(5)^{2}+40=25 + 40=65^{\circ}\), \(m\angle D=10\times5+35=50 + 35 = 85^{\circ}\), \(m\angle EFG=30\times5=150^{\circ}\), and \(65^{\circ}+85^{\circ}=150^{\circ}\) (valid)

Answer:

\(m\angle EFG = 150^{\circ}\)