Exercise 22 Page 379

Consider the given triangle.

We want to find the values of the variables x and y. 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 has two congruent angles, it appears that triangle LPM is an isosceles triangle. Next, we can use the Triangle Sum Theorem to find m∠ L.

m ∠ M+ m ∠ L+ m ∠ P=180

SubstituteValues

Substitute values

60+m∠ L+60=180

AddTerms

Add terms

m∠ L+120=180

SubEqn

LHS-120=RHS-120

m∠ L=60

We found that △ LMP is equiangular. Now we shall look at one of the Properties of Equilateral Triangles.

Properties of Equilateral Triangles
A triangle is equilateral if and only if it is equiangular.

Since our triangle is equiangular, it is actually an equilateral triangle. This means that all sides of the triangle are congruent as well. Then, we can setup two equations to find x and y. Equation I:&& 5x-3&=4 Equation II:&& 2y+6&=4 We will start with finding the value of x.

5x-3=4

AddEqn

LHS+3=RHS+3

5x=7

DivEqn

.LHS /5.=.RHS /5.

x=7/5

Next, we will find y.

2y+6=4

SubEqn

LHS-6=RHS-6

2y=-2

DivEqn

.LHS /2.=.RHS /2.

y=-1

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