Exercise 15 Page 379

Consider the given triangle.

We want to find the measure of angle BAC. To do so, we will start by identifying the type of triangle. Let's recall the classification of triangles.

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 and two base angles with the same measure.
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^(∘).

Since the given triangle have two congruent sides, it appears that triangle LPM is an isosceles triangle. Now we can consider the Isosceles Triangle Theorem to find m ∠ BAC.

Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Using this theorem, let's show the congruent angles and label them as x^(∘).

Now, we can find m ∠ BAC by using the Triangle Angle-Sum Theorem.

m ∠ BAC+m ∠ ACB+m ∠ CBA=180

SubstituteValues

Substitute values

x+x+60=180

AddTerms

Add terms

2x+60=180

SubEqn

LHS-60=RHS-60

2x=120

DivEqn

.LHS /2.=.RHS /2.

x=60

We found that the value of x is 60, so the measure of ∠ BAC is 60^(∘). Incidentally, the triangle is equiangular, this means that the triangle is also an equilateral triangle!

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