Question

solve for x.
in △def,
m∠d = 59°,
m∠e = 85°, and
m∠f = 4x°.
what is the value of x?
in △jkl,
∠j = (3x + 6)°,
∠k = (4x - 18)°, and
∠l = 5x°.
what is the value of x and what are the measures of ∠j, ∠k and ∠l?
solve for x.

Explanation:

Question 17

Step1: Use the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(x + 55=108\).

Step2: Solve for \(x\)

Subtract 55 from both sides of the equation: \(x=108 - 55\).

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(59+85 + 4x=180\).

Step2: Simplify the left - hand side

\(144+4x=180\).

Step3: Solve for \(x\)

Subtract 144 from both sides: \(4x=180 - 144=36\). Then divide both sides by 4: \(x=\frac{36}{4}=9\).

Step1: Use the triangle angle - sum theorem

\(\angle J+\angle K+\angle L = 180^{\circ}\). Substitute \(\angle J=(3x + 6)^{\circ}\), \(\angle K=(4x-18)^{\circ}\), and \(\angle L = 5x^{\circ}\) into the equation: \((3x + 6)+(4x-18)+5x=180\).

Step2: Combine like terms

\((3x+4x + 5x)+(6 - 18)=180\), \(12x-12 = 180\).

Step3: Solve for \(x\)

Add 12 to both sides: \(12x=180 + 12=192\). Divide both sides by 12: \(x = 16\).

Step4: Find the measures of the angles

\(\angle J=(3x + 6)^{\circ}=(3\times16+6)^{\circ}=54^{\circ}\).
\(\angle K=(4x-18)^{\circ}=(4\times16-18)^{\circ}=46^{\circ}\).
\(\angle L=5x^{\circ}=5\times16^{\circ}=80^{\circ}\).

Answer:

\(x = 53\)

Question 18