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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

4. Proving Angle Relationships

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Exercise 12 Page 305

Are ∠ 3 and ∠ 4 vertical angles? If so, what can you get from this?

Measures: m∠ 3=m∠ 4=113
Theorem: Vertical Angles Theorem.

Practice makes perfect

We are given the following system of equations. m∠ 3=2x+23 & (I) m∠ 4=5x-112 & (II) But also, from the diagram below we can see that ∠ 3 and ∠ 4 are vertical angles.

Hence, using the Vertical Angles Theorem, we obtain that ∠ 3 ≅ ∠ 4 and then, m∠ 3=m∠ 4. Using this fact, we can equate the equations in the given system and solve it for x.

m∠ 3=2x+23 & (I) m∠ 4=5x-112 & (II)

(I): m∠ 3= m∠ 4

m∠ 4=2x+23 m∠ 4=5x-112

▼

(II): Solve by substitution

(II): m∠ 4= 2x+23

m∠ 4=2x+23 2x+23=5x-112

(II): LHS-2x=RHS-2x

m∠ 4=2x+23 23=3x-112

(II): LHS+112=RHS+112

m∠ 4=2x+23 135=3x

(II): Rearrange equation

m∠ 4=2x+23 3x=135

(II): .LHS /3.=.RHS /3.

m∠ 4=2x+23 x=45

Now that we found the value of x, let's substitute it into the first equation to find m∠ 4.

m∠ 4=2x+23 x=45

(I): x= 45

m∠ 4=2( 45)+23 x=45

(I): Multiply

m∠ 4=90+23 x=45

(I): Add terms

m∠ 4=113 x=45

Finally, remember that ∠ 3≅ ∠ 4, which implies that m∠ 3=113 as well.

Subchapter links

Exercises p.304-307