Question

classify the triangle shown by its angles. justify your reasoning.
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: Use the triangle angle - sum property

The sum of the measures of the three angles of a triangle is \(180^{\circ}\). So, \((15x + 1)+(6x + 5)+(4x-1)=180\).
Combine like terms: \((15x+6x + 4x)+(1 + 5-1)=180\), which simplifies to \(25x+5 = 180\).
Subtract 5 from both sides: \(25x=180 - 5=175\).
Divide both sides by 25: \(x=\frac{175}{25}=7\).

Step2: Find the angle measures

Substitute \(x = 7\) into the expressions for each angle:

• For \((15x + 1)^{\circ}\): \(15\times7+1=105 + 1=106^{\circ}\)
• For \((6x + 5)^{\circ}\): \(6\times7+5=42+5 = 47^{\circ}\)
• For \((4x-1)^{\circ}\): \(4\times7-1=28 - 1=27^{\circ}\)

Step3: Classify the triangle

Because the triangle has an obtuse (\(>90^{\circ}\)) angle (\(106^{\circ}\)), it is an obtuse triangle.

Answer:

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