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, \(x + y=w\).

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 \(w-y=x>0\). So, \(w > y\).

Step3: 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 \(w - x=y>0\). So, \(w>x\).

Step4: Analyze option C (\(x + y=z\))

In a triangle, the sum of an interior angle and its adjacent exterior angle is \(180^{\circ}\), so \(z + w=180^{\circ}\), and \(x + y+z = 180^{\circ}\) (sum of interior angles of a triangle). But from the Exterior Angle Theorem \(x + y=w\), so \(w+z = 180^{\circ}\), which means \(x + y
eq z\) (unless \(w = z\) and \(w=90^{\circ}\), which is not a general case).

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

We know that \(x + y+z=180^{\circ}\) and \(x + y = w\), so \(w+z=180^{\circ}\). We also know that \(w=x + y\). But we cannot be sure that \(z>x\) in general. For example, if \(x = 60^{\circ}\), \(y=30^{\circ}\), then \(w = 90^{\circ}\) and \(z=90^{\circ}\), here \(z=x\).

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

From \(x + y=w\) and \(x + y+z=180^{\circ}\), we have \(w+z = 180^{\circ}\), so \(y + z
eq w\) (unless \(x = 180^{\circ}- 2z\), which is not a general case).

Answer:

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