Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ D E=24, E F=18, \mathrm{m} \angle D=15 $$

Step-by-Step Solution

Answer

One possible triangle with angles: 15°, 20°, and 145°.

Step 1: Identify the problem type

This exercise involves determining the number of possible triangles given two sides and a non-included angle, which is a situation for using the Law of Sines and checking for possible congruence conditions.

Step 2: Apply the Law of Sines

Use the Law of Sines to find possible angles. Given: \( DE = 24 \), \( EF = 18 \), \( m \angle D = 15^{\circ} \). By the Law of Sines, \( \frac{\sin \angle E}{24} = \frac{\sin 15^{\circ}}{18} \). Solve for \( \sin \angle E \): \( \sin \angle E = \frac{24 \times \sin 15^{\circ}}{18} \).

Step 3: Calculate sin(15 degrees)

Use the known value \( \sin 15^{\circ} \approx 0.2588 \) to compute \( \sin \angle E \). Substituting: \( \sin \angle E = \frac{24 \times 0.2588}{18} = 0.3442 \).

Step 4: Determine possible angles for \( \angle E \)

Check if \( \sin \angle E = 0.3442 \) gives valid angles. Calculate \( \angle E = \arcsin(0.3442) \approx 20^{\circ} \). Also, the supplement \( 180^{\circ} - 20^{\circ} = 160^{\circ} \) because \( \sin(\theta) = \sin(180^{\circ} - \theta) \). However, add up to see if triangle conditions permit.

Step 5: Check for triangle validity with both scenarios

1st scenario: \( \angle D = 15^{\circ}, \angle E = 20^{\circ} \), \( \angle F = 180^{\circ} - (15^{\circ} + 20^{\circ}) = 145^{\circ} \).2nd scenario: Invalid as \( \angle E = 160^{\circ} \) leaves \( 5^{\circ} \) for \( \angle F \), forming an unworkable large triangle sum.

Step 6: Conclusion about triangle

There is only one possible triangle with angles: \( \angle D = 15^{\circ}, \angle E = 20^{\circ}, \angle F = 145^{\circ} \).

Key Concepts

Triangle Congruence

Triangle congruence means that triangles are identical in shape and size, with corresponding sides and angles being equal. To determine congruence, you often use criteria such as ASA (Angle-Side-Angle), SSS (Side-Side-Side), or SAS (Side-Angle-Side). In problems where less information is known, like having two sides and a non-included angle, the ambiguity often requires checking specific congruence laws, such as the Law of Sines. This is crucial here, as the given information does not immediately ensure congruence. Hence, exploring all potential triangles using those measurements is necessary. Understanding these congruence concepts helps determine how many distinct triangles—if any—can be formed with the provided side lengths and angles.

Trigonometric Functions

Trigonometric functions are essential in solving for unknown angles or sides in triangles. The Law of Sines is particularly beneficial in cases with known angles and sides. In this exercise, you apply it to find an unknown angle, given by the formula:

• \( \frac{\sin \angle E}{24} = \frac{\sin 15^{\circ}}{18} \)

Given that we know \( \sin 15^{\circ} \) approximates to 0.2588, you can calculate \( \sin \angle E \). This results in \( 0.3442 \), a crucial part for determining possible angles through inverse trigonometric functions. This approach shows how interrelated trigonometric functions are within triangle problem-solving.

Triangle Validity

A triangle's validity hinges on the angles summing to 180 degrees. Not meeting this criterion means a valid triangle cannot be formed. Initially, we find the potential values for \( \angle E \), which are approximately 20° and its supplementary angle 160°. Here, testing for validity involves verifying whether the sum of these angles with the given Angle D creates a feasible triangle. In one case (\( \angle E = 20^{\circ} \)), an angle remained for \( \angle F = 145^{\circ} \) was feasible. But conversely, a 160° angle in the second scenario invalidates the triangle, leaving no meaningful sum for \( \angle F \). This checks how essential validation of sums is in ensuring triangles' feasibility.

Angle Calculation

Once you establish a valid triangle can exist, calculating remaining angles is crucial. Given two known angles, you can always find the third by subtracting their total from 180°. From our example:

• If \( \angle D = 15^{\circ} \) and \( \angle E = 20^{\circ} \), then \( \angle F = 180^{\circ} - (15^{\circ} + 20^{\circ}) = 145^{\circ} \).

These calculations confirm the triangle completes the angle sum property. Ensuring precise calculations when finding unknowns solidifies your understanding of resolving angles within the triangle by using basic arithmetic, complementing the trigonometric strategies used in initial steps.