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Pearson Geometry Common Core, 2011

PG

Pearson Geometry Common Core, 2011 View details

6. Proving Angles Congruent

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Exercise 35 Page 127

If angles are vertical, then they are congruent.

x=50, y=20
4y^(∘) = 80
2x^(∘) = 100
(x+y+10)^(∘) = 80

Practice makes perfect

To solve for x and y, we will need a system of equations. Notice that the labeled angles below are vertical angles. This means that they are congruent by the Vertical Angles Theorem.

We can equate the expressions for the two angle measures to write the first equation. 4y= x+y+10

Let's look at the diagram once more. This time including the third angle.

Because the third angle forms a linear pair with both of the other angles, we can create an equation using the fact that 2x added to either of these will equal 180^(∘). Let's use 4y. 2x+ 4y=180 Now that we have two equations, we write a system of equations. 4y=x+y+10 & (I) 2x+4y=180 & (II) We will solve it using the Substitution Method since x has a coefficient of 1 in Equation (I). After isolating x, we can substitute its value into Equation (II) to solve for y.

4y=x+y+10 2x+4y=180

(I): LHS-y=RHS-y

3y=x+10 2x+4y=180

(I): LHS-10=RHS-10

3y-10=x 2x+4y=180

(II): x= 3y-10

3y-10=x 2( 3y-10)+4y=180

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(II): Solve for y

(II): Distribute 2

3y-10=x 6y-20+4y=180

(II): Add terms

3y-10=x 10y-20=180

(II): LHS+20=RHS+20

3y-10=x 10y=200

(II): .LHS /10.=.RHS /10.

3y-10=x y=20

Now, we can substitute y=20 into Equation (I) to find x.

3y-10=x y=20

(I): y= 20

3( 20)-10=x y=20

▼

(I): Solve for x

(I): Multiply

60-10=x y=20

(I): Subtract term

50=x y=20

(I): Rearrange equation

x=50 y=20

Having found the values of both variables, we can calculate the measure of each angle.

Angle	x=50, y=20	Measure
4y	4( 20)	80
2x	2( 50)	100
x+y+10	50+ 20+10	80

Let's summarize our findings. &x=50, y=20 &4y^(∘) = 80 &2x^(∘) = 100 &(x+y+10)^(∘) = 80

Subchapter links

Lesson Check p.124

Standardized Test Prep p.127

Mixed Review p.127