Exercise 103 Page 201

Find the measures of the other two angles in △ CPM using vertical angles and the Alternate Exterior Angles Theorem.

∠ CPM=60^(∘)

Practice makes perfect

Examining the diagram, we can identify a couple of relationships.

The angles ∠ CMP and ∠ AMB are vertical angles. This means they are congruent according to the Vertical Angles Theorem.
The angles ∠ MCP and ∠ MAB are Alternate Interior angles. Since CP and AB are parallel segments, these angles are congruent according to the Alternate Interior Angles Theorem.

Let's mark this information in the diagram.

Congruent angles have the same measure which means we can set the angle's expressions equal to each other. Therefore, we can write two equations and If we combine them, we get a system of equations. 2x+3y = 3x-y 4x-3y= 65^(∘) To solve the system, we will use the Substitution Method. Solve the first equation for x and substitute it into the second one.

2x+3y=3x-y & (I) 4x-3y=65^(∘) & (II)

▼

(I): Solve for x

SubEqn

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

3y=x-y 4x-3y=65

RearrangeEqn

(I):Rearrange equation

x-y=3y 4x-3y=65

AddEqn

(I):LHS+y=RHS+y

x=4y 4x-3y=65

Substitute

(II):x= 4y

x=4y 4( 4y)-3y=65

▼

(II): Solve for y

Multiply

(II):Multiply

x=4y 16y-3y=65

SubTerm

(II):Subtract term

x=4y 13y=65

DivEqn

(II):.LHS /13.=.RHS /13.

x=4y y=5

Having solved for y, we substitute this back into the first equation to calculate x.

x=4y y=5

Substitute

(I):y= 5

x=4( 5) y=5

Multiply

(I):Multiply

x=20 y=5

When we know that x= 20^(∘) and y= 5^(∘), we can calculate ∠ MCP and ∠ CMP. ∠ MCP = 4( 20^(∘)) - 3( 5^(∘)) = 65^(∘) ∠ CMP = 2( 20^(∘)) + 3( 5^(∘)) = 55^(∘) Now we know two angles in △ CPM.

Finally, we want to find ∠ CPM. By the Triangle Angle Sum Theorem, the sum of the angles in △ CPM equals 180^(∘). Therefore, we can write an equation with the angles in △ CPM and solve for ∠ CPM.

∠ CPM + 55^(∘) + 65^(∘) = 180^(∘)

AddTerms

Add terms

∠ CPM + 120^(∘) = 180^(∘)

SubEqn

LHS-120^(∘)=RHS-120^(∘)

∠ CPM = 60^(∘)