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McGraw Hill Glencoe Geometry, 2012

MH

McGraw Hill Glencoe Geometry, 2012 View details

6. Isosceles and Equilateral Triangles

Finish the assignment

Continue to next subchapter

Exercise 39 Page 291

Start by considering the Isosceles Triangle Theorem, then use the Triangle Sum Theorem.

y=14

Practice makes perfect

Consider the given triangle.

We want to find the value of 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 sides, the given triangle is an isosceles triangle. Then, we can consider the Isosceles Triangle Theorem to find y.

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

Using this theorem, let's label the congruent angles.

Now we can find y by using the Triangle Sum Theorem.

(y^2-62)+(2y-5)+(2y-5)=180

Remove parentheses

y^2-62+2y-5+2y-5=180

Add and subtract terms

y^2+4y-72=180

LHS-180=RHS-180

y^2+4y-252=0

Write as a difference

y^2+18y-14y-252=0

Factor out y

y(y+18)-14y-252=0

Factor out -14

y(y+18)-14(y+18)=0

Factor out (y+18)

(y+18)(y-14)=0

Zero Property of Multiplication

lcy+18=0 & (I) y-14=0 & (II)

(I): LHS-18=RHS-18

ly=-18 y-14=0

(II): LHS+14=RHS+14

ly=-18 y=14

As a result, the value of y will be either -18 or 14. However, -18 may not satisfy 2y-5 because it is negative. Let's check!

2y-5

y= -18

2( -18)-5

Multiply

-36-5

Subtract term

-41

Since an angle measure cannot be negative, the only possible value of y is 14.

Subchapter links

Check Your Understanding p.289

Practice and Problem Solving p.289-292

H.O.T. Problems p.292

Standardized Test Practice p.293

Spiral Review p.293

Skills Review p.293