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Challenge

Investigating Triangles and Their Properties

To strengthen roof trusses, usually triangular shaped structures are used. In the diagram, $A C = C F,$ $A B = B D,$ and $D E = E F .$ The beams $\overline{B C},$ $\overline{D F},$ $\overline{C E},$ and $\overline{A D}$ are built in a way that $\overline{B C} ∥ \overline{D F}$ and $\overline{C E} ∥ \overline{A D} .$

Knowing that

\overline{C E}

is

43

inches long and

\overline{D F}

is

68

inches long, what would be the lengths of

\overline{A D}

and

\overline{B C} ?

Discussion

Properties of a Triangle's Interior Angles

Considering the previous exploration, the sum of interior angles of a triangle can be derived.

Rule

Interior Angles Theorem

The sum of the measures of the interior angles of a triangle is

18 0^{\circ} .

The triangle ABC with movable vertices A, B, and C and the angle measures written

Based on this diagram, the following relation holds true.

$m ∠ A + m ∠ B + m ∠ C = 18 0^{\circ}$

This theorem is also known as the Triangle Angle Sum Theorem.

Proof

Consider a triangle with vertices $A,$ $B,$ and $C,$ and the parallel line to $\overline{B C}$ through $A .$ Let $∠ 1$ and $∠ 2$ be the angles outside $△ A B C$ formed by this line and the sides $\overline{A B}$ and $\overline{A C} .$

By the Alternate Interior Angles Theorem, $∠ B$ is congruent to $∠ 1$ and $∠ C$ is congruent to $∠ 2 .$

By the definition of congruent angles,

∠ 1

and

∠ B

have the same measure. For the same reason,

∠ 2

and

∠ C

also have the same measure.

∠ B ≅ ∠ 1 ⇕ m ∠ B = m ∠ 1 ∠ C ≅ ∠ 2 ⇕ m ∠ C = m ∠ 2

Furthermore, in the diagram it can be seen that

∠ B A C,

∠ 1,

and

∠ 2

form a straight angle. Therefore, by the Angle Addition Postulate their measures add to

18 0^{\circ} .

m ∠ B A C + m ∠ 1 + m ∠ 2 = 18 0^{\circ}

By the Substitution Property of Equality, the sum of the measures of

∠ B A C,

∠ B,

and

∠ C

is equal to

18 0^{\circ} .

m ∠ B A C + m ∠ 1 + m ∠ 2 = 18 0^{\circ} ⇓ m ∠ B A C + m ∠ B + m ∠ C = 18 0^{\circ}

Finally, in

△ A B C,

∠ B A C

can be named

∠ A .

m ∠ B A C + m ∠ B + m ∠ C = 18 0^{\circ} ⇓ m ∠ A + m ∠ B + m ∠ C = 18 0^{\circ}

Example

Solving Problems Using the Interior Angles Theorem

Dylan is designing a wooden sofa made of oak wood for his local park. The sides of the sofa will have identical dimensions in the shape of a triangle. He already has decided on the angle measures of the top corner and bottom-right corner of each side.

To cut the sides of the sofa out of the board using a table saw, which can cut at angles, Dylan needs to find the measure of the third angle. Dylan's hands are full — help him find the measure of the third angle.

Hint

Use the Interior Angles Theorem.

Solution

Since the sides have a triangular shape and the measures of two angles are known, the Interior Angles Theorem can be used to find the missing angle measure. Let $x$ be the measure of the missing angle.

3 5^{\circ} + 4 0^{\circ} + x = 18 0^{\circ}

Solving this equation for $x,$ the measure of the missing angle can be found.

3 5^{\circ} + 4 0^{\circ} + x = 18 0^{\circ}

Add terms

7 5^{\circ} + x = 18 0^{\circ}

$LHS - 7 5^{\circ} = RHS - 7 5^{\circ}$

x = 10 5^{\circ}

Discussion

Properties of a Triangle's Exterior Angles

The previous exploration shows that there is a clear relation between an exterior angle of a triangle and its remote interior angles.

Example

Solving Problems Using the Triangle Exterior Angles Theorem

Dylan is almost ready to cut the sides of the sofa. Before doing so, he wants to be sure that people sitting on his sofa can lean back freely and feel comfortable. Therefore, he needs to find the measure of the angle exterior to the third angle.

Note that if the angle measure is less than

9 0^{\circ},

the sofa is inclined backwards. Dylan is a bit busy with handling the wood. Help him find the measure of the exterior angle.

Hint

Use the Triangle Exterior Angle Theorem.

Solution

Recall that by the Triangle Exterior Angle Theorem, the measure of a triangle's exterior angle is equal to the sum of the measures of its two remote interior angles. Therefore, the measure of the exterior angle $x$ can be expressed.

x = 3 5^{\circ} + 4 0^{\circ} ⇓ x = 7 5^{\circ}

Since the measure of the exterior angle is less than $9 0^{\circ},$ the people sitting on this sofa can lean back and feel comfortable. Thanks for helping Dylan.

Explore

Investigating Properties of Isosceles Triangles

Given that

△ A B C

is a right triangle, reflect it across

\overline{A B} .

Examine the triangle formed by the preimage and image. Compare the base angles of the triangle.

Discussion

Properties of Isosceles Triangles

Reflecting a right triangle about either of its legs forms an isosceles triangle. Note that a reflection is a rigid motion, so the side lengths and the interior angles of the right triangle are preserved.

Rule

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent.

An isosceles triangle.

Based on this diagram, the following relation holds true.

$\overline{A B} ≅ \overline{A C}$ $\Rightarrow$ $∠ B ≅ ∠ C$

The Isosceles Triangle Theorem is also known as the Base Angles Theorem.

Proof

Geometric Approach

Consider a triangle $A B C$ with two congruent sides, or an isosceles triangle.

In this triangle, let

P

be the point of intersection of

\overline{B C}

and the angle bisector of

∠ A .

From the diagram, the following facts about

△ B A P

and

△ C A P

can be observed.

Statement	Reason
$∠ B A P ≅ ∠ C A P$	Definition of an angle bisector
$\overline{B A} ≅ \overline{C A}$	Given
$\overline{A P} ≅ \overline{A P}$	Reflexive Property of Congruence

Therefore,

△ B A P

and

△ C A P

have two pairs of corresponding congruent sides and one pair of congruent included angles. By the Side-Angle-Side Congruence Theorem,

△ B A P

and

△ C A P

are congruent triangles.

△ B A P ≅ △ C A P

Corresponding parts of congruent figures are congruent. Therefore,

∠ B

and

∠ C

are congruent.

∠ B ≅ ∠ C

It has been proven that if two sides of a triangle are congruent, then the angles opposite them are congruent.

Proof

Using Transformations

Consider an isosceles triangle $△ A B C .$

A line passing through $A$ and the midpoint of $\overline{B C}$ will be drawn. Let $P$ be the midpoint.

Since $\overline{B P}$ and $\overline{P C}$ are congruent, the distance between $B$ and $P$ is equal to the distance between $C$ and $P .$ Therefore, $B$ is the image of $C$ after a reflection across $A P .$ Also, because $A$ lies on $A P,$ a reflection across $A P$ maps $A$ onto itself. The same is true for $P .$

Reflection Across $A P$
Preimage	Image
$C$	$B$
$A$	$A$
$P$	$P$

The table shows that the images of the vertices of

△ C A P

are the vertices of

△ B A P .

It can be concluded that

△ B A P

is the image of

△ C A P

after a reflection across

A P .

Since a reflection is a rigid motion, this proves that the triangles are congruent.

Corresponding parts of congruent figures are congruent, so

∠ B

and

∠ C

are congruent.

∠ B ≅ ∠ C

Example

Solving Problems Using the Isosceles Triangle Theorem

Dylan notices that he needs a support beam to support the seat. The bottoms of each side panel are $3$ feet long. Therefore, if he places the support beam from the corner with the larger angle measure to the opposite side in a position where the endpoint of the support beam is $3$ feet away from the bottom-right corner, then it will fit just right.

In this case, what should be the measure of the angle between the support beam and the bottom of the side panel?

Hint

Consider the Base Angles Theorem.

Solution

Placing the support beam as shown forms an isosceles triangle.

Recall that according to the Base Angles Theorem, base angles of an isosceles triangle are congruent. It can be seen that the measure of the vertex angle is

4 0^{\circ} .

Assuming that the measure of a base angle of the triangle is

x,

an equation can be written by the Interior Angles Theorem.

x + x + 4 0^{\circ} = 18 0^{\circ}

By solving this equation, the measure of the angle between the support beam and the bottom of the side can be found.

x + x + 4 0^{\circ} = 18 0^{\circ}

Add terms

2 x + 4 0^{\circ} = 18 0^{\circ}

$LHS - 4 0^{\circ} = RHS - 4 0^{\circ}$

2 x = 14 0^{\circ}

$LHS / 2 = RHS / 2$

x = 7 0^{\circ}

Explore

Investigating Properties of a Triangle's Midsegment

In the following applet, investigate the rigid motions by moving the slider.

What is the resulting figure formed by the preimage and images? What is the relationship between

\overline{B A^{''}}

and

\overline{A C^{'''}} ?

Discussion

Properties of a Triangle's Midsegment

As it is seen in the previous exploration, using the rigid motions, the Triangle Midsegment Theorem can be proven.

Example

Solving Problems Using the Triangle Midsegment Theorem

Finally, Dylan is ready to place the seat. He plans to place it just above the support beam such that it will be parallel to the bottom. Therefore, the corners of the seat will be at the midpoints of the sides.

How can he find the width of the seat knowing that the bottom of the side is $3$ feet long.

Hint

The seat will be aligned with the midsegment of the triangular side.

Solution

Since the corners of the seat are at the midpoints of the triangular side, it will be aligned with the midsegment of the triangular side. Therefore, by the Triangle Midsegment Theorem, the width of the seat will be half the length of the bottom of the side.

\frac{3}{2} = 1.5 ft

The width of the seat is

1.5

feet.

Discussion

Solving Problems Using a Triangle's Properties

In this lesson, the investigated theorems about triangles have been proven using a variety of methods. Furthermore, with the help of these theorems, the challenge provided at the beginning of the lesson can be solved. Recall the diagram.

Consider the given information about the beams of the roof.

∙ A C = C F ∙ A B = B D ∙ D E = E F ∙ \overline{B C} ∥ \overline{D F} ∙ \overline{C E} ∥ \overline{A D}

From here, what are the lengths of

\overline{A D}

and

\overline{B C} ?

Hint

Use the Triangle Midsegment Theorem.

Solution

By the definition of a midsegment, both $\overline{B C}$ and $\overline{C E}$ are midsegments of $△ A D F .$ By the Triangle Midsegment Theorem, $C E$ is half of $A D,$ and $B C$ is half of $D F .$

C E B C = \frac{1}{2} A D = \frac{1}{2} D F

Knowing that $C E$ is $43$ inches and $D F$ is $68$ inches, these values can be substitute into these equations to find $A D$ and $B C .$

43 B C = \frac{1}{2} A D \Rightarrow A D = 86 = \frac{1}{2} (68) \Rightarrow B C = 34

Therefore, the length of $\overline{A D}$ is $86$ inches and the length of $\overline{B C}$ is $34$ inches.

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