Question

consider triangle wxy. wx is 5, xy is 10, wy is 14. which statement about the angles is true? ○ angle w is greater than angle y. ○ angle y is the largest angle. ○ angle x is smaller than angle w. ○ angle w is the smallest angle.

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. Let's first identify the sides of triangle \(WXY\). The sides are: \(WX = 5\), \(XY=10\), and \(WY = 14\).

Now, let's identify the angles opposite these sides:

• The side \(WX = 5\) is opposite angle \(Y\).
• The side \(XY = 10\) is opposite angle \(W\).
• The side \(WY=14\) is opposite angle \(X\).

Step2: Compare the lengths of the sides

We have the side lengths in the order: \(5<10 < 14\), i.e., \(WX

Step3: Relate side lengths to angle measures

Since the side opposite an angle determines the size of the angle (larger side opposite larger angle), we can say:

• The angle opposite \(WX\) (angle \(Y\)) is smaller than the angle opposite \(XY\) (angle \(W\)) because \(WX
• The angle opposite \(WY\) (angle \(X\)) is the largest angle because \(WY\) is the longest side.

Now let's analyze each option:

• Option 1: "Angle \(W\) is greater than angle \(Y\)". Since \(WX = 5\) (opposite \(\angle Y\)) and \(XY = 10\) (opposite \(\angle W\)) and \(5 < 10\), by the angle - side relationship, \(\angle Y<\angle W\), so this statement is true. But let's check other options to be sure.
• Option 2: "Angle \(Y\) is the largest angle". The longest side is \(WY = 14\), which is opposite \(\angle X\), so \(\angle X\) is the largest angle, not \(\angle Y\). So this is false.
• Option 3: "Angle \(X\) is smaller than angle \(W\)". The longest side \(WY = 14\) is opposite \(\angle X\), so \(\angle X\) is the largest angle, so \(\angle X>\angle W\). This is false.
• Option 4: "Angle \(W\) is the smallest angle". The smallest side is \(WX = 5\), which is opposite \(\angle Y\), so \(\angle Y\) is the smallest angle, not \(\angle W\). This is false.

Answer:

Angle W is greater than angle Y.