Question

question 1 (4 points)
solve for x. then find the measure of the angles.

x =

m∠d =

m∠e =

m∠f =

blank 1:

blank 2:

blank 3:

blank 4:

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
For \(\triangle DEF\), \((11x - 21)^{\circ}=(5x - 2)^{\circ}+(9x + 3)^{\circ}\)

$$11x-21=(5x - 2)+(9x + 3)$$

Step2: Simplify the right - hand side

$$11x-21=5x - 2+9x + 3$$

$$11x-21 = 14x+1$$

Step3: Solve for \(x\)

Subtract \(11x\) from both sides:
\(-21=3x + 1\)
Subtract \(1\) from both sides:
\(-22 = 3x\)

$$x = 8$$

Step4: Find \(m\angle D\)

Substitute \(x = 8\) into \(m\angle D=(5x - 2)^{\circ}\)

$$m\angle D=(5\times8 - 2)^{\circ}=(40 - 2)^{\circ}=38^{\circ}$$

Step5: Find \(m\angle E\)

Substitute \(x = 8\) into \(m\angle E=(9x + 3)^{\circ}\)

$$m\angle E=(9\times8+3)^{\circ}=(72 + 3)^{\circ}=75^{\circ}$$

Step6: Find \(m\angle F\)

Substitute \(x = 8\) into \(m\angle F=(11x - 21)^{\circ}\)

$$m\angle F=(11\times8-21)^{\circ}=(88 - 21)^{\circ}=67^{\circ}$$

Answer:

Blank 1: \(8\)
Blank 2: \(38^{\circ}\)
Blank 3: \(75^{\circ}\)
Blank 4: \(67^{\circ}\)