Question

compare the given measures by choosing < , > , or = .
if there is not enough information, select \cannot be determined\.
if necessary, you may learn what the markings on a figure indicate.

(image of three geometric figures with comparison questions:)

1. triangle wxy and wyz: wx > wz (selected), or cannot be determined.
2. triangles tuv and def: compare m∠t and m∠e (select option), or cannot be determined (selected).
3. triangle abc with segment bd: compare m∠abd and m∠cbd (select option), or cannot be determined (selected).

\try again\ button visible.

Explanation:

Step1: Analyze the third triangle (ABD and CBD)

In triangle \(ABD\) and \(CBD\), we have \(AB = CB = 6\), \(AD = 8\), \(CD = 9\). We use the Hinge Theorem (SAS Inequality Theorem), which states that if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is shorter than the third side of the second triangle, then the included angle of the first triangle is smaller than the included angle of the second triangle.

For \(\triangle ABD\) and \(\triangle CBD\):

• \(AB = CB = 6\) (given)
• \(BD\) is common to both triangles.
• \(AD = 8\) and \(CD = 9\), so \(AD < CD\).

By the Hinge Theorem, the angle opposite the shorter side is smaller. The angle opposite \(AD\) is \(\angle ABD\) and the angle opposite \(CD\) is \(\angle CBD\). Since \(AD < CD\), we have \(m\angle ABD < m\angle CBD\).

Step2: Correct the previous selection

The previous selection was "Cannot be determined", but using the Hinge Theorem, we can determine the relationship. So we should select \(<\) for \(m\angle ABD\) and \(m\angle CBD\).

Answer:

\(m\angle ABD < m\angle CBD\) (select \(<\) in the dropdown for \(m\angle ABD\) and \(m\angle CBD\))