Question

10,362 Learners found this answer helpful

list the angle measures of δbcd in order from smallest to largest. assume that t is a positive number.

(image of triangle bcd with sides: bd = 27t, cd = 34t, bc = 37t)

m∠□ < m∠□ < m∠□

Explanation:

Step1: Recall the triangle side - angle relationship

In a triangle, the larger the length of a side, the larger the measure of the opposite angle. This is known as the "larger side - larger opposite angle" theorem.

Step2: Identify the side lengths and their opposite angles

• Side \( BD = 27t \), its opposite angle is \( \angle C \).
• Side \( CD = 34t \), its opposite angle is \( \angle B \).
• Side \( BC = 37t \), its opposite angle is \( \angle D \).

Step3: Compare the side lengths

Since \( 27t<34t < 37t \) (because \( t>0 \), we can compare the coefficients directly), according to the side - angle relationship:

• The side \( BD = 27t \) is the shortest, so its opposite angle \( \angle C \) is the smallest.
• The side \( CD = 34t \) is longer than \( BD \) but shorter than \( BC \), so its opposite angle \( \angle B \) is larger than \( \angle C \) but smaller than \( \angle D \).
• The side \( BC = 37t \) is the longest, so its opposite angle \( \angle D \) is the largest.

Answer:

\( m\angle C