Question

1. fill in the blanks to complete the paragraph proof of the exterior angle theorem.
2. practice in the diagram below,

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

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that \(m\angle EFG=m\angle E + m\angle D\). So, \((30x)=(x^{2}+40)+(10x + 35)\).

Step2: Simplify the equation

$$
LATEXBLOCK0
$$

Step3: Factor the quadratic equation

For \(x^{2}-20x + 75=0\), we look for two numbers that multiply to \(75\) and add up to \(- 20\). The numbers are \(-15\) and \(-5\). So, \((x - 15)(x - 5)=0\).

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 the values of \(x\)

• When \(x = 15\):

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

• When \(x = 5\):

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

Answer:

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