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Congruence, Proof, and Constructions

Theorems About Triangles

In the previous lesson, some theorems about lines and angles have been explored and proven. This lesson will focus on some theorems about triangles.

Catch-Up and Review

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

Challenge

Investigating Triangles and Their Properties

To strengthen roof trusses, usually triangular shaped structures are used. In the diagram, and The beams and are built in a way that and

Knowing that is inches long and is inches long, what would be the lengths of and

Explore

Investigating a Triangle's Interior Angles

Consider where and are the midpoints of and respectively. Perform two rotations on — one about point and the other about point
Considering the images and preimage, investigate the sum of the interior angles of

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 interior angles of a triangle is

Based on this diagram, the following relation holds true.

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

Proof

Consider a triangle with vertices and and the parallel line to through Let and be the angles outside formed by this line and the sides and

triangle

By the Alternate Interior Angles Theorem, is congruent to and is congruent to

triangle

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

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

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 be the measure of the missing angle.

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

Explore

Investigating a Triangle's Exterior Angles

Once again, consider where and are the midpoints of and respectively. This time, begin by rotating about Then, rotate the resulting figure about
Considering the images and preimage, what can be concluded about exterior angle and its remote interior angles and

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.

Theory

Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Triangle with an exterior angle marked

Based on the diagram above, the following relation holds true.

Proof

Using Properties of Angles

Consider a triangle with vertices and with the exterior angle corresponding to

Triangle with an exterior angle marked

The diagram shows that and form a linear pair. Therefore, the sum of their measures is Additionally, by the Triangle Angle Sum Theorem, the sum of the measures of the angles of is

Now can be isolated in Equation (I). Next, the expression of can be substituted into Equation (II).
Solve for
It has been proven that the measure of is equal to the sum of the measures of and Therefore, it can be said that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Proof

Using Transformations

Consider where and are the midpoints of and respectively. Let be the exterior angle of

Triangle with an exterior angle marked
Now, can be rotated over Since a rotation is a rigid motion, the image of after the rotation is congruent to Corresponding parts of congruent figures are congruent, so the measures of the angles and the lengths of the sides remain unchanged.
Triangle interior angles included side rotation
Since a rotation is equivalent to a reflection, is parallel to and is parallel to Therefore, is a parallelogram and is congruent to Now the parallelogram will be rotated over
Triangle exterior angle parallelogram rotation
By the Parallelogram Opposite Angles Theorem, is congruent to Congruent angles have the same measure by the definition. Since is equal to the sum of and is equal to the sum of and because of the Transitive Property of Equality.

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

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 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.

Explore

Investigating Properties of Isosceles Triangles

Given that is a right triangle, reflect it across
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.

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

Proof

Geometric Approach

Consider a triangle with two congruent sides, or an isosceles triangle.

An isosceles triangle.
In this triangle, let be the point of intersection of and the angle bisector of
An isosceles triangle with an angle bisector.
From the diagram, the following facts about and can be observed.
Statement Reason
Definition of an angle bisector
Given
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.

Proof

Using Transformations

Assume is an isosceles triangle.

An isosceles triangle.

A line passing through and the midpoint of will be drawn. Let be the midpoint.

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

Reflection Across
Preimage Image
The above table shows that the images of the vertices of are the vertices of Therefore, is the image of after a reflection across Since a reflection is a rigid motion, this proves that the triangles are congruent.
An isosceles triangle.
Corresponding parts of congruent figures are congruent, so and are congruent.

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 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?

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 Assuming that the measure of a base angle of the triangle is an equation can be written by the Interior Angles Theorem. By solving this equation, the measure of the angle between the support beam and the bottom of the side can be found.

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 and

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.

Rule

Triangle Midsegment Theorem

The line segment that connects the midpoints of two sides of a triangle — also known as a midsegment — is parallel to the third side of the triangle and half its length.
If is a midsegment of then the following statement holds true.

and

Proof

Using Coordinates

This theorem can be proven by placing the triangle on a coordinate plane. For simplicity, vertex will be placed at the origin and vertex on the axis.

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.

Segment Endpoints Substitute Simplify
and
and

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.

Segment Endpoints Length Simplify
and
and

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.

Proof

Using Transformations
This proof will be developed based on the given diagram, but it is valid for any triangle.
Movable triangle with a midsegment
To prove this theorem, it must be proven that is parallel to and is equal to half Each statement will be proven one at a time.

This part can be proven by using rigid motions. Translate along so that is mapped onto Since is the midpoint of is mapped onto
Translation of the triangle in a triangle with a midsegment
Next, it must be proven that the image of lies on This proof will be done using indirect reasoning.

1

Assume That Does Not Lie on

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

2

Show That

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.

Recall that Additionally, is the common side of and By the Side-Side-Side Congruence Theorem, is congruent to Because corresponding angles of congruent triangles are congruent, is congruent to

Since translations preserve angles, is parallel to By the Alternate Interior Angles Theorem, is congruent to Therefore, by the Transitive Property of Congruence,

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

3

Contradiction

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

It is now proven that lies on Because translations preserve angles, is congruent to

By the Converse Corresponding Angles Theorem, is parallel to

Since and are parallel, another rigid motion to will be applied.

1

Rotate Around the Midpoint of
Rotate counterclockwise about the midpoint of so that is mapped onto It can be noted that the triangle is rotated Since rotations preserve angles and lengths, this rotation maps onto Therefore, is mapped onto
The Midsegment Theorem proof first rotation

2

Rotate Around the Midpoint of
Rotate counterclockwise about the midpoint of so that is mapped onto It can be noted that the triangle is rotated Since rotations preserve angles and lengths, this rotation maps onto Therefore, and are mapped onto and respectively.
The Midsegment Theorem proof second rotation
By the Segment Addition Postulate, the length of can be calculated by adding the lengths of smaller segments. Because corresponding sides of congruent triangles are congruent, is congruent to and is congruent to Therefore, and By the Substitution Property of Equality, can be expressed in terms of Finally, by the Division Property of Equality, the second statement of the theorem is obtained.

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 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. The width of the seat is 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.

From here, what are the lengths of and

Solution

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.

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