Angles and Transversals - Geometry in Two Dimensions (Pre-Algebra)

Interior Angles	Exterior Angles
$∠ 2,$ $∠ 4,$ $∠ 6,$ $∠ 8$	$∠ 1,$ $∠ 3,$ $∠ 5,$ $∠ 7$

Interior Angles

Exterior Angles

∠ 2,

∠ 4,

∠ 6,

∠ 8

∠ 1,

∠ 3,

∠ 5,

∠ 7

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 3 ≅ ∠ 5$ and $∠ 4 ≅ ∠ 6$
$∠ 3 ≅ ∠ 5$ or $∠ 4 ≅ ∠ 6$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 3 ≅ ∠ 5

and

∠ 4 ≅ ∠ 6

∠ 3 ≅ ∠ 5

∠ 4 ≅ ∠ 6

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 7$ and $∠ 2 ≅ ∠ 8$
$∠ 1 ≅ ∠ 7$ or $∠ 2 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 7

and

∠ 2 ≅ ∠ 8

∠ 1 ≅ ∠ 7

∠ 2 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ and $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$
$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ or $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

and

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ and $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$
$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ or $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

and

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

Consider what I drew on Zimbabwe's flag. The number of countries in all of Africa equals three times $x .$ How many countries are there in Africa?

Pair	Position Relative to the Vertex	Position Relative to the Transversal
$∠ 1$ and $∠ 5$	Northeast	Right
$∠ 2$ and $∠ 6$	Northwest	Left
$∠ 3$ and $∠ 7$	Southwest	Left
$∠ 4$ and $∠ 8$	Southeast	Right

Pair

Position Relative to the Vertex

Position Relative to the Transversal

∠ 1

and

∠ 5

Northeast

Right

∠ 2

and

∠ 6

Northwest

Left

∠ 3

and

∠ 7

Southwest

Left

∠ 4

and

∠ 8

Southeast

Right

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ and $∠ 4 ≅ ∠ 8$
$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ or $∠ 4 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

and

∠ 4 ≅ ∠ 8

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

∠ 4 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

a Try to find a relationship about the given angles to find the measure of the lower angle. It is a good idea to begin by drawing the sides of each angle.

Notice that the vertical sides lie on the left vertical beam. That implies they lie on the same line. This line acts like a transversal cutting the two parallel diagonal beams.

Draw the given information without the photo in the background for a clearer visual of the angles, lines, and transversal.

The diagram shows that the angles lie on the same side of the transversal. One of them is an exterior angle while the other is an interior angle. This means that these angles are corresponding angles. The angles have the same measure because the lines are parallel.

(2 x + 6)^{\circ} = (4 x - 28)^{\circ}

Next, solve the equation for

x .

2 x + 6 = 4 x - 28

SubEqn

$LHS - 2 x = RHS - 2 x$

6 = 2 x - 28

AddEqn

$LHS + 28 = RHS + 28$

34 = 2 x

DivEqn

$LHS / 2 = RHS / 2$

17 = x

RearrangeEqn

Rearrange equation

x = 17

Now that the value of

x

is known, substitute it into the measure of the lower angle.

(4 x - 28)^{\circ}

Substitute

$x = 17$

(4 \cdot 17 - 28)^{\circ}

Multiply

(68 - 28)^{\circ}

SubTerm

Subtract term

4 0^{\circ}

The measure of the angle labeled below the bridge is

4 0^{\circ} .

Notice that the angle labeled above the bridge also measures

4 0^{\circ} .

b Start by drawing the lines containing the sides of the angles labeled.

The two diagonal bars are parallel to each other according to what Dylan sees. The upper angle is exterior while the lower angle is an interior angle. Additionally, both angles lie on the same side of the transversal.

Those characteristics suggest that these two angles are corresponding angles. As a result, they have the same measure.

(6 y + 2)^{\circ} = (7 (y - 1) - 3)^{\circ}

Next, solve the equation for

y .

6 y + 2 = 7 (y - 1) - 3

Distr

Distribute $7$

6 y + 2 = 7 y - 7 - 3

SubTerm

Subtract term

6 y + 2 = 7 y - 10

SubEqn

$LHS - 6 y = RHS - 6 y$

2 = y - 10

AddEqn

$LHS + 10 = RHS + 10$

12 = y

RearrangeEqn

Rearrange equation

y = 12

Finally, substitute

12

for

y

into the expression that represents the measure of the upper angle.

(6 y + 2)^{\circ}

Substitute

$y = 12$

(6 \cdot 12 + 2)^{\circ}

Multiply

(72 + 2)^{\circ}

AddTerms

Add terms

7 4^{\circ}

The measure of the upper angle, and also the measure of the lower angle, is

7 4^{\circ} .

Angles	Relation
$∠ 7$ and $∠ 4$	Alternate Interior Angles
$∠ 7$ and $∠ 3$	Consecutive Interior Angles
$∠ 7$ and $∠ 2$	Corresponding Angles
$∠ 7$ and $∠ 5$	Vertical Angles

Angles

Relation

∠ 7

and

∠ 4

Alternate Interior Angles

∠ 7

and

∠ 3

Consecutive Interior Angles

∠ 7

and

∠ 2

Corresponding Angles

∠ 7

and

∠ 5

Vertical Angles

Angles	Relation	Relationship Between Measures
$∠ 7$ and $∠ 4$	Alternate Interior Angles	$Congruent$
$∠ 7$ and $∠ 3$	Consecutive Interior Angles	$Supplementary$
$∠ 7$ and $∠ 2$	Corresponding Angles	$Congruent$
$∠ 7$ and $∠ 5$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 7

and

∠ 4

Alternate Interior Angles

Congruent

∠ 7

and

∠ 3

Consecutive Interior Angles

Supplementary

∠ 7

and

∠ 2

Corresponding Angles

Congruent

∠ 7

and

∠ 5

Vertical Angles

Congruent

Angles	Relation	Relationship Between Measures
$∠ 1$ and $∠ 8$	Alternate Exterior Angles	$Congruent$
$∠ 1$ and $∠ 5$	Same-Side Exterior Angles	$Supplementary$
$∠ 1$ and $∠ 6$	Corresponding Angles	$Congruent$
$∠ 1$ and $∠ 3$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 1

and

∠ 8

Alternate Exterior Angles

Congruent

∠ 1

and

∠ 5

Same-Side Exterior Angles

Supplementary

∠ 1

and

∠ 6

Corresponding Angles

Congruent

∠ 1

and

∠ 3

Vertical Angles

Congruent

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Catch-Up and Review

Fixing the Fence

Line Intersecting Two or More Lines

Transversal

Angles Formed by a Transversal

Interior and Exterior Angles

Relationships Between Interior Angles

Alternate Interior Angles

The Louvre Museum

Hint

Solution

Extra

Relationships Between Exterior Angles

Alternate Exterior Angles

La Plata City - Argentina

Hint

Solution

Interior Angles on the Same Side of the Transversal

Same-Side Interior Angles

Pyramids of Giza

Hint

Solution

Extra

Exterior Angles on the Same Side of the Transversal

Same-Side Exterior Angles

Number of Countries in Africa

Hint

Solution

Relating Interior and Exterior Angles

Corresponding Angles

Bridges and Boats

Hint

Solution

Extra

Identifying Pairs of Angles

Parallel Lines and Transversals

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