Question

1. triangle a is an isosceles triangle with two angles of measure x degrees and one angle of measure y degrees.

a. find three combinations of x and y that make this sentence true.
b. write an equation relating x and y.
c. if you were to sketch the graph of this linear equation, what would its slope be? how can you interpret the slope in the context of the triangle?
(from unit 3, lesson 13.)

1. consider the following graphs of linear equations. decide which line has a positive slope, and which has a negative slope. then calculate each line’s exact slope.

Explanation:

Part a

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). For an isosceles triangle with two angles \(x\) and one angle \(y\), we have \(2x + y=180\). We can solve for \(y = 180 - 2x\) or \(x=\frac{180 - y}{2}\). We need to find values of \(x\) and \(y\) such that all angles are positive (greater than \(0^\circ\)) and the triangle inequality for angles holds (each angle \(< 180^\circ\)).

Step2: Find first combination

Let \(x = 50\). Then \(y=180 - 2\times50=180 - 100 = 80\). Check: \(2\times50+80 = 180\), and all angles \(50,50,80\) are positive and less than \(180\).

Step3: Find second combination

Let \(x = 60\). Then \(y = 180-2\times60=180 - 120 = 60\). Check: \(2\times60 + 60=180\), angles \(60,60,60\) (equilateral, a special isosceles triangle) are valid.

Step4: Find third combination

Let \(x = 80\). Then \(y=180 - 2\times80=180 - 160 = 20\). Check: \(2\times80+20 = 180\), angles \(80,80,20\) are valid.

Step1: Use triangle angle sum

The sum of the interior angles of a triangle is \(180^\circ\). In an isosceles triangle with two angles of measure \(x\) and one angle of measure \(y\), the sum of the angles is \(x + x+ y=180\).

Step2: Simplify the equation

Combine like terms: \(2x + y=180\). We can also rearrange it as \(y=180 - 2x\) or \(x=\frac{180 - y}{2}\).

Step1: Rewrite the equation in slope - intercept form

We have the equation \(y = 180-2x\). The slope - intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step2: Identify the slope

Comparing \(y=- 2x + 180\) with \(y = mx + b\), we see that the slope \(m=-2\).

Step3: Interpret the slope

The slope of \(-2\) means that for every \(1\) - degree increase in the measure of the equal angles (\(x\)), the measure of the third angle (\(y\)) decreases by \(2\) degrees. This is because the equation \(y = 180-2x\) shows a linear relationship where \(y\) is a function of \(x\) with a rate of change of \(- 2\).

Answer:

(for part a):
Three combinations are \((x = 50,y = 80)\), \((x = 60,y = 60)\), \((x = 80,y = 20)\) (answers may vary as long as \(2x + y=180\) and \(x>0,y>0\))

Part b