Question

question 10 of 10
which of the following statements must be true about this diagram? check
all that apply.

a. x + y = w
b. w > y
c. x + y = z
d. w > x
e. z > x
f. y + z = w

Explanation:

Step1: Recall the Exterior Angle Theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In the given triangle, the exterior angle is \(w^{\circ}\), and the two non - adjacent interior angles are \(x^{\circ}\) and \(y^{\circ}\). So, by the exterior angle theorem, \(x + y=w\). This makes option A correct.

Step2: Analyze Option B (\(w>y\))

From \(x + y=w\) (where \(x>0\) because it is an angle in a triangle), we can rewrite it as \(w - y=x\). Since \(x>0\), then \(w-y>0\), which implies \(w > y\). So option B is correct.

Step3: Analyze Option C (\(x + y=z\))

We know that \(x + y+z = 180^{\circ}\) (sum of angles in a triangle) and \(w+z=180^{\circ}\) (linear pair). From \(w + z=180^{\circ}\), we have \(z = 180 - w\). From \(x + y=w\), we have \(z=180-(x + y)\). So \(x + y
eq z\) (unless \(w = 90^{\circ}\), but this is not a general case), so option C is incorrect.

Step4: Analyze Option D (\(w>x\))

From \(x + y=w\) (where \(y>0\) because it is an angle in a triangle), we can rewrite it as \(w - x=y\). Since \(y>0\), then \(w - x>0\), which implies \(w>x\). So option D is correct.

Step5: Analyze Option E (\(z>x\))

We know that \(x + y+z=180\) and \(w=x + y\), \(w+z = 180\). We cannot be sure that \(z>x\) in general. For example, if \(x = 60^{\circ}\), \(y = 30^{\circ}\), then \(w=90^{\circ}\) and \(z = 90^{\circ}\), so \(z=x\) in this case (if \(x = 60\) and \(z = 90\), no, wait, if \(x=60\), \(y = 30\), \(w = 90\), then \(z=90\), so \(z>x\) here. But if \(x = 80^{\circ}\), \(y=10^{\circ}\), then \(w = 90^{\circ}\), \(z=90^{\circ}\), \(zx\). Take \(x = 90\), \(y = 30\), then \(w = 120\), \(z=60\), here \(zx\) always, so option E is incorrect.

Step6: Analyze Option F (\(y + z=w\))

From \(x + y=w\) and \(x + y+z=180\), we have \(w+z = 180\), so \(y+z=180 - x
eq w\) (unless \(x = 0\), which is not possible), so option F is incorrect.

Answer:

A. \(x + y = w\), B. \(w>y\), D. \(w>x\)