Question

directions: complete the following questions by showing all work and annotations. keep work organized and box final answer. all work must be shown in order to receive full credit. #1.) in the diagram below ∠n is a right - angle, m∠o=(x²)° and m∠m=(40 + 5x)°. a.) find the value of x. b.) determine m∠m. c.) classify the triangle. self assessment how do you feel about today’s lesson? what questions do you still have?

Explanation:

Step1: Use angle - sum property of triangle

The sum of the interior angles of a triangle is 180°. Since ∠N = 90°, we have the equation \(x^{2}+(40 + 5x)+90=180\).
\[x^{2}+5x + 40+90=180\]
\[x^{2}+5x+130 = 180\]
\[x^{2}+5x - 50=0\]

Step2: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 5\), \(c=-50\)), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor. Factoring \(x^{2}+5x - 50=(x + 10)(x - 5)=0\).
Setting each factor equal to zero gives \(x+10 = 0\) or \(x - 5=0\). So \(x=-10\) or \(x = 5\). But since an angle measure cannot be negative when considering \(x\) in the context of angle measures, we take \(x = 5\).

Step3: Find m∠M

Substitute \(x = 5\) into the expression for m∠M.
m∠M=(40 + 5x)°.
m∠M=40+5×5=40 + 25=65°.

Step4: Find m∠O

Substitute \(x = 5\) into the expression for m∠O.
m∠O=x^{2}. So m∠O = 5^{2}=25°.

Step5: Classify the triangle

Since one angle (∠N) is 90°, the triangle is a right - triangle. And since all three angles (25°, 65°, 90°) are different, it is a scalene right - triangle.

Answer:

a. \(x = 5\)
b. m∠M = 65°
c. Scalene right - triangle