Question

classify the triangle shown by its angles. justify your reasoning.

(15x + 1)°
(6x + 5)° (4x − 1)°

select choice the sum of the measures of the three angles of a triangle is select choice. so, (15x + 1) + (6x + 5) + (4x − 1) = select choice and x = select choice. substituting select choice into the expressions for each angle, the angle measures are select choice. because the triangle has an select choice angle, it is select choice.

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, set the sum of the three angles equal to \(180\):
\((15x + 1)+(6x + 5)+(4x - 1)=180\)

Step2: Simplify the left side

Combine like terms:
\(15x + 6x + 4x + 1 + 5 - 1 = 25x + 5\)
So, \(25x + 5 = 180\)

Step3: Solve for \(x\)

Subtract \(5\) from both sides:
\(25x = 180 - 5 = 175\)
Divide by \(25\):
\(x=\frac{175}{25}=7\)

Step4: Find each angle

• First angle: \(15x + 1 = 15(7)+1 = 106^\circ\)
• Second angle: \(6x + 5 = 6(7)+5 = 47^\circ\)
• Third angle: \(4x - 1 = 4(7)-1 = 27^\circ\)

Step5: Classify the triangle

Since one angle (\(106^\circ\)) is greater than \(90^\circ\), it is an obtuse angle. A triangle with an obtuse angle is an obtuse triangle.

Answer:

• The sum of the measures of the three angles of a triangle is \(\boldsymbol{180^\circ}\).
• So, \((15x + 1)+(6x + 5)+(4x - 1)=\boldsymbol{180}\) and \(x = \boldsymbol{7}\).
• Substituting \(\boldsymbol{7}\) into the expressions for each angle, the angle measures are \(\boldsymbol{106^\circ}\), \(\boldsymbol{47^\circ}\), \(\boldsymbol{27^\circ}\).
• Because the triangle has an \(\boldsymbol{obtuse}\) (greater than \(90^\circ\)) angle, it is an \(\boldsymbol{obtuse}\) triangle.