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.
prove: ( mangle efg=mangle e + mangle d )
linear pair triangle sum theorem right angle substitution
( 180^{circ}) ( mangle efg=mangle e + mangle d ) angle addition postulate ( mangle edf )
( mangle efg + mangle efd=) because the angles form a
( mangle e + mangle d + mangle efd=)
because of the. so, by,
( mangle efg + mangle efd=mangle e + mangle d + mangle efd ). by the subtraction property of equality,
.

1. practice in the diagram below,

( mangle efg=(30x)^{circ},mangle e=(x^{2}+40)^{circ} ), and
( mangle d=(10x + 35)^{circ} ). find the measure of ( angle efg ).

Explanation:

Step1: Use the Exterior Angle Theorem

According to the Exterior Angle Theorem \(m\angle EFG=m\angle E + m\angle D\). Substitute the given expressions: \((30x)^{\circ}=(x^{2}+40)^{\circ}+(10x + 35)^{\circ}\).

Step2: Simplify the equation

$$
LATEXBLOCK0
$$

Step3: Factor the quadratic equation

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

Step4: Solve for \(x\)

Set each factor equal to zero: \(x-15 = 0\) gives \(x = 15\); \(x - 5=0\) gives \(x = 5\).

Step5: Check for valid solutions

If \(x = 5\): \(m\angle EFG=(30\times5)^{\circ}=150^{\circ}\), \(m\angle E=(5^{2}+40)^{\circ}=65^{\circ}\), \(m\angle D=(10\times5 + 35)^{\circ}=85^{\circ}\), and \(65+85 = 150\).
If \(x = 15\): \(m\angle E=(15^{2}+40)^{\circ}=265^{\circ}\) (invalid as angle measure in a triangle cannot be more than \(180^{\circ}\)).

Answer:

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