Exercise 46 Page 253

Let's start by looking at the angles of the given triangle as labeled by Curtis.

His friend Arnoldo says that at least one of his measures is incorrect. We want to find two different ways to explain why Arnoldo knows that this is true.

First Explanation

The Triangle Angle-Sum Theorem tells us that the measures of the interior angles of a triangle add up to 180^(∘). Let's check whether this is true or not for the measurements that Curtis wrote down.

Since the measurements do not add up to 180^(∘), they cannot all be correct.

Second Explanation

Let's classify the angles of the triangle according to Curtis's measures.

According to Corollary 4.2 of the Triangle Angle-Sum Theorem, there can only be at most one obtuse angle in any given triangle. Since there are two obtuse angle measures among Curtis's measurements, they cannot be both correct.

Extra

Types of Angles

Angles can be classified by their measures. For angles between 0^(∘) and 180^(∘), they can be divided into four categories: acute, right, obtuse, and straight. Acute angle &: measures less than 90 ^(∘) Right angle &: measures exactly 90 ^(∘) Obtuse angle &: measures greater than 90 ^(∘) Straight angle &: measures exactly 180 ^(∘)