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Suppose a triangle has vertices A

, B and C and has the following measurements: ¯¯¯¯¯¯¯¯AB=23.3 cm¯¯¯¯¯¯¯¯AC=87.6 cm¯¯¯¯¯¯¯¯BC=65.2 cm

Start by drawing a diagram of this triangle and labeling the known values.

What is the degree measure of ∠C

?

m∠C=

To find the degree measure of ∠C, we can use the **Law of Cosines**.

Given the sides of the triangle:
- \( AB = 23.3 \) cm
- \( AC = 87.6 \) cm
- \( BC = 65.2 \) cm

### Law of Cosines Formula:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( c = AC = 87.6 \) cm (opposite side of ∠C)
- \( a = AB = 23.3 \) cm
- \( b = BC = 65.2 \) cm

We need to find \( \angle C \). Rearrange the formula to solve for \( \cos(C) \):

\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Substituting the known values:

\[
\cos(C) = \frac{(23.3)^2 + (65.2)^2 - (87.6)^2}{2 \times 23.3 \times 65.2}
\]

Now, I'll calculate this expression to determine the measure of \( \angle C \).The degree measure of \( \angle C \) is approximately \( 161.41^\circ \).

This indicates that \( \angle C \) is an obtuse angle, as expected given the lengths of the triangle's sides.

Would you like more details, or do you have any other questions?

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Here are 8 related questions that can expand on this concept:

1. How would you determine \( \angle A \) using the Law of Cosines?
2. How can the Law of Sines be applied after finding \( \angle C \)?
3. What is the sum of all angles in any triangle?
4. How can we check if the given side lengths form a valid triangle?
5. What is the relationship between side lengths and angle types in a triangle?
6. How does the Law of Cosines differ from the Law of Sines?
7. How can the area of the triangle be calculated using these sides?
8. How would you solve for \( \angle C \) if you only had the perimeter of the triangle?

**Tip:** Remember that the Law of Cosines is particularly useful when dealing with non-right triangles or when you know all sides and need to find an angle.

Discover how to find the degree measure of angle C in a triangle with sides AB = 23.3 cm, AC = 87.6 cm, and BC = 65.2 cm using the Law of Cosines. Learn step-by-step calculations and understand the concept of obtuse angles in triangles. Suitable for Grades 10-12 students studying geometry and trigonometry.

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