Exercise 5 Page 378

Consider the Converse of Isosceles Triangle Theorem.

z=12

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Consider the given triangle and its measures.

In the given figure, we can see that m∠ TSR is congruent to m∠ RTS. Let's identify the type of the given triangle by recalling 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 the base angles are congruent.
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 base angles of the given triangles are congruent, we can said that triangle RST is a isosceles triangle. Then, we can consider the Converse of Isosceles Triangle Theorem to find the variable z. The theorem states the following.

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

Using the theorem, we can conclude that RS≅ RT. This means that both segments have the same length. Then, we can equate them and write an equation that helps us find z. RS = RT Let's substitute RS = 2z-15 and RT= 9 in the above equation and solve it for z.

RS = RT

SubstituteII

RS= 2z-15, RT= 9

2z-15= 9

AddEqn

LHS+15=RHS+15

2z=24

DivEqn

.LHS /2.=.RHS /2.

z=12

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