Exercise 50 Page 342

Let's draw a diagram and label the angles at A.

We know that adjacent exterior and interior angles form a straight angle, meaning that their measures add up to 180^(∘). m∠ 1+ m∠ A=180^(∘) This equation can be rearranged to express the measure of the interior angle in terms of the measure of the corresponding exterior angle. m∠ A=180^(∘)- m∠ 1 If the exterior angle is acute, then its measure is less than 90^(∘). Let's see what this means for the measure of the interior angle.

m∠ 1<90^(∘)

MultNegIneq

Multiply by -1 and flip inequality sign

- m∠ 1>- 90^(∘)

AddIneq

LHS+180^(∘)>RHS+180^(∘)

180^(∘)-m∠ 1>180^(∘)- 90^(∘)

SubTerms

Subtract terms

180^(∘)-m∠ 1>90^(∘)

Substitute

180^(∘)-m∠ 1= m∠ A

m∠ A>90^(∘)

Since any angle measure in a triangle is less that 180^(∘), this means that angle ∠ A is obtuse. Since the triangle has an obtuse angle, it is an obtuse triangle.

Extra

Classification of Triangles

Triangles can be classified either according to their side lengths or to their internal angle measures.

The following table lists all the different types of triangles according to their corresponding classification.

Classification of Triangles
Scalene Triangle	A scalene triangle is a triangle in which all three sides have different lengths.
Isosceles Triangle	An isosceles triangle is a triangle that has two congruent sides.
Equilateral Triangle	An equilateral triangle is a triangle in which all the sides are congruent.
Acute Triangle	An acute triangle is a triangle where all angles are less than 90^(∘) or π2.
Obtuse Triangle	An obtuse triangle is a triangle with exactly one an angle whose measure is greater than 90^(∘) or π2.
Right Triangle	A right triangle is a specific type of triangle that contains one angle of 90^(∘).

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