Question

directions: complete the following questions by showing all work and annotations. keep work organized and box any final answer. all work must be shown in order to receive full credit.
**#1.) in △abc, m∠a=(4x + 5)°, m∠b=(x + 80)°, and m∠c=(5x - 5)°. what is the type of triangle is △abc?
a) equilateral
b) isosceles
c) right scalene
d) right isosceles
2.) given the triangle below, classify it using the best name and then solve for all angles.
self - assessment: how would you rate your understanding of todays lesson?
what question(s) do you still have about this lesson?

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A+m\angle B + m\angle C=180^{\circ}\).
Substitute \(m\angle A=(4x + 5)^{\circ}\), \(m\angle B=(x + 80)^{\circ}\), and \(m\angle C=(5x-5)^{\circ}\) into the equation:
\((4x + 5)+(x + 80)+(5x-5)=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms:
\((4x+x + 5x)+(5 + 80-5)=180\)
\(10x+80 = 180\)

Step3: Solve for \(x\)

Subtract 80 from both sides:
\(10x=180 - 80\)
\(10x=100\)
Divide both sides by 10:
\(x = 10\)

Step4: Find the measures of the angles

• For \(m\angle A=(4x + 5)^{\circ}\), substitute \(x = 10\):

\(m\angle A=(4\times10+5)^{\circ}=45^{\circ}\)

• For \(m\angle B=(x + 80)^{\circ}\), substitute \(x = 10\):

\(m\angle B=(10 + 80)^{\circ}=90^{\circ}\)

• For \(m\angle C=(5x-5)^{\circ}\), substitute \(x = 10\):

\(m\angle C=(5\times10-5)^{\circ}=45^{\circ}\)

Since two angles (\(m\angle A = 45^{\circ}\) and \(m\angle C=45^{\circ}\)) are equal and one angle (\(m\angle B = 90^{\circ}\)) is a right angle.

Answer:

d) Right Isosceles