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To strengthen roof trusses, usually triangular shaped structures are used. In the diagram, $AC=CF,$ $AB=BD,$ and $DE=EF.$ The beams $BC,$ $DF,$ $CE,$ and $AD$ are built in a way that $BC∥DF$ and $CE∥AD.$

Knowing that $CE$ is $43$ inches long and $DF$ is $68$ inches long, what would be the lengths of $AD$ and $BC?$Considering the previous exploration, the sum of interior angles of a triangle can be derived.

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.

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Use the Interior Angles Theorem.

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.

$35_{∘}+40_{∘}+x=180_{∘} $

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

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

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?

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Consider the Base Angles Theorem.

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 $40_{∘}.$ 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+40_{∘}=180_{∘} $

By solving this equation, the measure of the angle between the support beam and the bottom of the side can be found.
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 $BA_{′′}$ and $AC_{′′′}?$

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

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.

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

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.

$23 =1.5ft $

The width of the seat is $1.5$ feet.
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.

$∙AC=CF∙AB=BD∙DE=EF ∙BC∥DF∙CE∥AD $

From here, what are the lengths of $AD$ and $BC?$ {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">D<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"inches","answer":{"text":["86"]}}

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Use the Triangle Midsegment Theorem.

By the definition of a midsegment, both $BC$ and $CE$ are midsegments of $△ADF.$ By the Triangle Midsegment Theorem, $CE$ is half of $AD,$ and $BC$ is half of $DF.$

$CEBC =21 AD=21 DF $

Knowing that $CE$ is $43$ inches and $DF$ is $68$ inches, these values can be substitute into these equations to find $AD$ and $BC.$

$43BC =21 AD⇒AD=86=21 (68)⇒BC=34 $

Therefore, the length of $AD$ is $86$ inches and the length of $BC$ is $34$ inches.