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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

2. Congruent Triangles

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Exercise 19 Page 349

Begin by using the Triangle Angle-Sum Theorem. Then, consider the Third Angles Theorem, to write a system of two equations.

x=4
y=2

Practice makes perfect

Let's add some labels to the given diagram.

Before we try to find the values of x and y, we will use the Triangle Angle-Sum Theorem to find the measure of ∠ A.

m∠ A+m∠ B+m∠ C=180

m∠ B= 148, m ∠ C= 18

m∠ A+ 148+ 18=180

Add terms

m∠ A+166=180

LHS-166=RHS-166

m∠ A=14

Looking at the angle markers, we know that ∠ A and ∠ E as well as ∠ B and ∠ G are congruent angles. This means that each pair of angles has equal measures. m∠ A = m∠ E m∠ B = m∠ G By the Third Angles Theorem, ∠ C and ∠ F also have the same measure. We can use these congruence relations to create equations that we can use to find the values of x and y. m∠ A = m∠ E ⇔ 14=3x+y m∠ C = m∠ F ⇔ 18=5x-y From here, we can create a system of two equations. 14=3x+y & (I) 18=5x-y & (II) To solve this system, we will use the Elimination Method.

14=3x+y & (I) 18=5x-y & (II)

(I): Add (II)

14+ 18=3x+y+( 5x-y) 18=5x-y

▼

(II):Solve for x

(I): Add and subtract terms

32=8x 18=5x-y

(I): .LHS /8.=.RHS /8.

4=x 18=5x-y

(I): Rearrange equation

x=4 18=5x-y

Now we can find the value of y by substituting the value of x into Equation (II).

x=4 18=5x-y

(II): x= 4

x=4 18=5( 4)-y

▼

(I):Solve for y

(II): Multiply

x=4 18=20-y

(II): LHS+y=RHS+y

x=4 y+18=20

(II): LHS-18=RHS-18

x=4 y=2

Subchapter links

Exercises p.347-352