Question

find the measure of the missing angle. * the measures of the angles of a triangle are shown in the figure below. solve for x. two sides of a triangle have lengths of 11 and 6. find the range of possible measures for the third side. give your answer as a compound inequality without spaces!

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, we have the equation \(54^{\circ}+90^{\circ}+(7x + 1)^{\circ}=180^{\circ}\).

Step2: Simplify the left - hand side of the equation

First, add the known angles: \(54 + 90=144\). The equation becomes \(144+(7x + 1)=180\). Then, combine like - terms: \(144 + 1+7x=180\), which simplifies to \(145+7x=180\).

Step3: Solve for \(x\)

Subtract 145 from both sides of the equation: \(7x=180 - 145\), so \(7x = 35\). Divide both sides by 7: \(x=\frac{35}{7}=5\).

Step4: Find the measure of the missing angle

Substitute \(x = 5\) into the expression for the missing angle \((7x + 1)^{\circ}\). We get \(7\times5+1=35 + 1=36^{\circ}\).

Answer:

The measure of the missing angle is \(36^{\circ}\) and \(x = 5\).