To strengthen roof trusses, usually triangular shaped structures are used. In the diagram, and The beams and are built in a way that and
Considering the previous exploration, the sum of interior angles of a triangle can be derived.
Based on this diagram, the following relation holds true.
This theorem is also known as the Triangle Angle Sum Theorem.
By the definition of congruent angles, and have the same measure. For the same reason, and also have the same measure. Furthermore, in the diagram it can be seen that and form a straight angle. Therefore, by the Angle Addition Postulate their measures add to By the Substitution Property of Equality, it follows that the sum of the measures of and is equal to Finally, in can be named
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.
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 be the measure of the missing angle.
Solving this equation for the measure of the missing angle can be found.
Based on the diagram above, the following relation holds true.
Consider a triangle with vertices and with the exterior angle corresponding to
Now can be isolated in Equation (I). Next, the expression of can be substituted into Equation (II).
Consider where and are the midpoints of and respectively. Let be the exterior angle of
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 the sofa is inclined backwards. Dylan is a bit busy with handling the wood. Help him find the measure of the exterior angle.
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 can be expressed.
Since the measure of the exterior angle is less than the people sitting on this sofa can lean back and feel comfortable. Thanks for helping Dylan.
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.
Based on this diagram, the following relation holds true.
The Isosceles Triangle Theorem is also known as the Base Angles Theorem.
Consider a triangle with two congruent sides, or an isosceles triangle.
|Definition of an angle bisector|
|Reflexive Property of Congruence|
Therefore, and have two pairs of corresponding congruent sides and one pair of congruent included angles. By the Side-Angle-Side Congruence Theorem, and are congruent triangles. Corresponding parts of congruent figures are congruent. Therefore, and are congruent. It has been proven that if two sides of a triangle are congruent, the angles opposite them are congruent.
Assume is an isosceles triangle.
Since and are congruent, the distance between and is equal to the distance between and Therefore, is the image of after a reflection across Also, because lies on a reflection across maps onto itself. The same is true for
Dylan notices that he needs a support beam to support the seat. The bottoms of each side panel are 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 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?
Placing the support beam as shown forms an isosceles triangle.
Since lies on the origin, its coordinates are Point is on the axis, meaning its coordinate is The remaining coordinates are unknown, and can be named and If is the midsegment from to then by definition of a midpoint, and are the midpoints of and respectively.
To prove this theorem, it must be proven that is parallel to and that is half
If the slopes of these two segments are equal, then they are parallel. The coordinate of both and is Therefore, is a horizontal segment. Next, the coordinates of and will be found using the Midpoint Formula.
The coordinate of both and is Therefore, is also a horizontal segment. Since all horizontal segments are parallel, it can be said that and are parallel.
Since both and are horizontal, their lengths are given by the difference of the coordinates of their endpoints.
Since is half of it can be stated that the midsegment is half the length of Therefore, a midsegment of a triangle is parallel to the third side of the triangle and half its length.
Assume that the image of after the translation along does not lie on Then lies either above or below The proof will be developed for only one case, but it is valid for both of them.
Based on the assumption, let denote the point of intersection of and
Now, it will be proven that is congruent to This will lead to the contradiction, because then would be equal to This is not possible, since lies between and It is given that because is the midpoint of Additionally, since is translated along it can be concluded that Then, by the Transitive Property of Equality.
Again, since translations preserve angles, is congruent to Additionally, is congruent to One more time, by the Transitive Property of Congruence,
It can also be noted that By the Angle-Side-Angle Congruence Theorem, is congruent to
Because corresponding sides of congruent triangles are congruent, it follows that is equal to This means that is equal to because is the midpoint of However, since lies between and it cannot be true that Therefore, this contradiction verifies that the image of must lie on
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 feet long.
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. The width of the seat is 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.
From here, what are the lengths of and
By the definition of a midsegment, both and are midsegments of By the Triangle Midsegment Theorem, is half of and is half of
Knowing that is inches and is inches, these values can be substitute into these equations to find and
Therefore, the length of is inches and the length of is inches.