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In this lesson, some theorems about lines and angles will be explored and proven. The theorems will be applied using real-life examples.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Parallel lines
Perpendicular lines
Vertical angles
Supplementary angles
Rigid motions
Corresponding angles
Alternate interior angles
Perpendicular bisector

Example

Solving Problems Using Vertical Angles

In Flowerland Village, there is a crossroad between Tulip Street and Rose Street. There is a plan to continue the construction of Tulip Street toward the southwest. At the moment, the crossroad forms two angles, whose measures are expressed by

3 x + 14

and

6 x - 5,

respectively.

A crossroad before and after the construction

Find the measures of all four angles the crossroad will form after the construction of Tulip Street is finished.

Hint

Use the given expressions to form an equation for $x .$ Identify the relationship between $∠ 1$ and $∠ 3,$ as well as $∠ 2$ and $∠ 4$ by analyzing their positions.

Solution

The angles formed by the crossroad before the construction form a linear pair. Therefore, they are supplementary angles. Using this information, the following equation can be formed.

6 x - 5 + 3 x + 14 = 18 0^{\circ}

By solving it, the value of

x

can be determined.

6 x - 5 + 3 x + 14 = 18 0^{\circ}

Solve for

x

Add and subtract terms

9 x + 9 = 18 0^{\circ}

$LHS - 9 = RHS - 9$

9 x = 17 1^{\circ}

$LHS / 9 = RHS / 9$

x = 1 9^{\circ}

Now that the value of

x

is known, the measures of

∠ 1

and

∠ 2

can be calculated.

$x = 19$
$m ∠ 1 = 3 x + 14$	$m ∠ 2 = 6 x - 5$
$m ∠ 1 = 3 (19) + 14$	$m ∠ 2 = 6 (19) - 5$
$m ∠ 1 = 57 + 14$	$m ∠ 2 = 114 - 5$
$m ∠ 1 = 7 1^{\circ}$	$m ∠ 2 = 10 9^{\circ}$

Next, by analyzing the position of

∠ 1

and

∠ 3,

as well as

∠ 2

and

∠ 4,

it can be noted that these are vertical angles. Therefore, by the Vertical Angles Theorem, they are two pairs of congruent angles.

m ∠ 1 = m ∠ 3 m ∠ 2 = m ∠ 4 \Rightarrow m ∠ 3 = 7 1^{\circ} m ∠ 4 = 10 9^{\circ}

In this way, it was obtained that

∠ 1

and

∠ 3

are each

7 1^{\circ},

and

∠ 2

and

∠ 4

are each

10 9^{\circ} .

Discussion

Corresponding Angles Theorem and Its Converse

The observed relation between corresponding angles is presented and proven in the following theorem.

Note that the converse statement is also true.

Example

Solving Problems Using Corresponding Angles

In a Flowerland Village house, there are stairs with hand railings like shown in the diagram. The measures of $∠ 1$ and $∠ 2$ are expressed as $5 t - 2$ and $4 t + 12,$ respectively.

Railings of the staircase

What are the measures of

∠ 1

and

∠ 2 ?

Hint

How do measures of $∠ 1$ and $∠ 2$ relate to each other? Use the given expressions to form an equation for $t .$

Solution

Analyzing the diagram, it can be noted that

∠ 1

and

∠ 2

are corresponding angles formed by two parallel lines and a transversal. Therefore, by the Corresponding Angles Theorem, these angles are congruent. Hence, the measures of these angles are the same.

∠ 1 m ∠ 1 ≅ ∠ 2 ⇓ = m ∠ 2

By substituting

m ∠ 1

with

5 t - 2

and

m ∠ 2

with

4 t + 12,

the equation for

t

can be formed.

m ∠ 1 = m ∠ 2

$m ∠ 1 = 5 t - 2$ , $m ∠ 2 = 4 t + 12$

5 t - 2 = 4 t + 12

Solve for

t

$LHS - 4 t = RHS - 4 t$

t - 2 = 12

$LHS + 2 = RHS + 2$

t = 14

Now that the value of

t

is known, the measure of each of the angles can be calculated.

m ∠ 1 = 5 t - 2

Substitute

14

for

t

and evaluate

$t = 14$

m ∠ 1 = 5 (14) - 2

Multiply

m ∠ 1 = 70 - 2

Subtract term

m ∠ 1 = 68

Since the angles are congruent, it can be concluded that they both measure to be

6 8^{\circ} .

m ∠ 1 = m ∠ 2 m ∠ 1 = 6 8^{\circ} \Rightarrow m ∠ 2 = 6 8^{\circ}

Discussion

Alternate Interior Angles Theorem and Its Converse

Like corresponding angles, alternate interior angles are also formed by two parallel lines cut by a transversal.

The converse statement is also true.

Discussion

Alternate Exterior Angles Theorem and Its Converse

Similar properties can be discovered for alternate exterior angles.

Example

Using Alternate Interior Angles to Solve Problems

In order to build Tulip Street on the south side of the Lilian river, which goes through Flowerland Village, there is a need to build a bridge. Devontay, an architect, proposed the following plan for the bridge.

A plan of a bridge

It is known that the measure of $∠ 1$ is equal to $4 a + 11$ and the measure of $∠ 2$ is equal to $8 a - 53 .$ What are the measures of $∠ 1$ and $∠ 2 ?$

Hint

How do the measures of $∠ 1$ and $∠ 2$ relate to each other? Use the given expressions to form an equation for $a .$

Solution

By analyzing the diagram it can be noted that $∠ 1$ and $∠ 2$ are alternate interior angles.

Therefore, by the Alternate Interior Angles Theorem, these angles are congruent. Hence, the measures of these angles are the same.

∠ 1 m ∠ 1 ≅ ∠ 2 ⇓ = m ∠ 2

By substituting

m ∠ 1

with

4 a + 11

and

m ∠ 2

with

8 a - 53,

the equation for

a

can be formed.

m ∠ 1 = m ∠ 2

$m ∠ 1 = 4 a + 11$ , $m ∠ 2 = 8 a - 53$

4 a + 11 = 8 a - 53

Solve for

a

$LHS - 4 a = RHS - 4 a$

11 = 4 a - 53

$LHS + 53 = RHS + 53$

64 = 4 a

$LHS / 4 = RHS / 4$

16 = a

Rearrange equation

a = 16

Knowing the value of

a,

the measure of each of these angles can be calculated.

m ∠ 1 = 4 a + 11

Substitute

16

for

a

and evaluate

$a = 16$

m ∠ 1 = 4 (16) + 11

Multiply

m ∠ 1 = 64 + 11

Add terms

m ∠ 1 = 75

Since the angles are congruent, it can be concluded that they both measure

7 5^{\circ} .

m ∠ 1 = m ∠ 2 m ∠ 1 = 7 5^{\circ} \Rightarrow m ∠ 2 = 7 5^{\circ}

Explore

Investigating Points on a Perpendicular Bisector

Up to now, some basic theorems about angles have been seen and proven through some rigid motions. Before the end of the lesson, one last theorem about segments will be learned. Consider a perpendicular bisector of a segment

\overline{A B} .

Move

C

and the endpoints of the segment and compare the distances

A C

and

B C .

Perpendicular bisector of AB

What conjecture can be made about position of

C

in respect to the endpoints

A

and

B

of the segment? Does this conjecture also apply to other points on the perpendicular bisector

\overline{C M} ?

Discussion

Perpendicular Bisector Theorem and Its Converse

Closure

Solving Problems Using Perpendicular Bisectors

In Flowerland Village, there are two related families, Funnystongs and Cleverstongs, who live opposite each other. Mr. Funnystong and Mr. Cleverstong want to pave a road between the houses so that every point of the road is equidistant to their houses.

The locating of two houses opposite to each other

If the houses are

16

meters away, how far from the houses and along what line should the road be paved?

Answer

Distance: $8$ meters from each house.
Direction: Along the perpendicular bisector to the segment with endpoints at the houses.

Hint

What does the the Perpendicular Bisector Theorem state?

Solution

Recall what the Perpendicular Bisector Theorem states.

Any point on a perpendicular bisector is equidistant from the endpoints of the line segment.

With this theorem in mind, the position of the road can be determined. To do so, draw a segment whose endpoints are located at the houses.

Before drawing the perpendicular bisector of this segment, its midpoint should be found. Since the distance between the houses is $16$ meters, the perpendicular bisector will pass through a point that is $\frac{1 6}{2} = 8$ meters away from the houses.

Based on the theorem, it can be said that each point on the bisector is equidistant from the houses. Therefore, the road between the houses should be paved along the segment's perpendicular bisector.

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