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In order to operate with triangles, the relationships between the angles and sides of a triangle need to be investigated. Therefore, in this lesson, some theorems about triangles will be built upon several theorems about lines and angles.
### Catch-Up and Review

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

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 $BC?$
Consider $△ABC,$ where $D$ and $E$ are the midpoints of $AB$ and $AC,$ respectively. Perform two $180_{∘}$ rotations on $△ABC$ — one about point $D$ and the other about point $E.$

Considering the images and preimage, investigate the sum of the interior angles of $△ABC.$

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

The sum of the measures of the interior angles of a triangle is $180_{∘}.$
### Proof

Based on this diagram, the following relation holds true.

$m∠A+m∠B+m∠C=180_{∘}$

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

Consider a triangle with vertices $A,$ $B,$ and $C,$ and the parallel line to $BC$ through $A.$ Let $∠1$ and $∠2$ be the angles outside $△ABC$ formed by this line and the sides $AB$ and $AC.$

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 $∠BAC,$ $∠1,$ and $∠2$ form a straight angle. Therefore, by the Angle Addition Postulate their measures add to $180_{∘}.$
$m∠BAC+m∠1+m∠2=180_{∘} $

By the Substitution Property of Equality, the sum of the measures of $∠BAC,$ $∠B,$ and $∠C$ is equal to $180_{∘}.$
$m∠BAC+m∠1+m∠2=180_{∘}⇓m∠BAC+m∠B+m∠C=180_{∘} $

Finally, in $△ABC,$ $∠BAC$ can be named $∠A.$ $m∠BAC+m∠B+m∠C=180_{∘}⇓m∠A+m∠B+m∠C=180_{∘} $

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.

Once again, consider $△ABC,$ where $D$ and $E$ are the midpoints of $AB$ and $AC,$ respectively. This time, begin by rotating $△ABC$ about $D.$ Then, rotate the resulting figure about $E.$

Considering the images and preimage, what can be concluded about exterior angle $∠ACF$ and its remote interior angles $∠A$ and $∠B?$

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

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles, or remote interior angles.
### Proof

Using Properties of Angles
It has been proven that the measure of $∠PCA$ is equal to the sum of the measures of $∠A$ and $∠B.$ 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

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

$m∠PCA=m∠A+m∠B$

Consider a triangle with vertices $A,$ $B,$ and $C,$ and one of the exterior angles corresponding to $∠C.$

The diagram shows that $∠C$ and $∠PCA$ form a linear pair, so the sum of their measures is $180_{∘}.$ Additionally, by the Triangle Angle Sum Theorem, the sum of the angle measures of $△ABC$ is $180_{∘}.$${m∠C+m∠PCA=180_{∘}m∠A+m∠B+m∠C=180_{∘} (I)(II) $

Now $m∠C$ can be isolated in Equation (I).
$m∠C+m∠PCA=180_{∘}⇕m∠C=180_{∘}−m∠PCA $

Next, the expression of $m∠C$ can be substituted into Equation (II).
$m∠A+m∠B+m∠C=180_{∘}$

Substitute

$m∠C=180_{∘}−m∠PCA$

$m∠A+m∠B+(180_{∘}−m∠PCA)=180_{∘}$

Solve for $m∠PCA$

RemovePar

Remove parentheses

$m∠A+m∠B+180_{∘}−m∠PCA=180_{∘}$

SubEqn

$LHS−180_{∘}=RHS−180_{∘}$

$m∠A+m∠B−m∠PCA=0$

AddEqn

$LHS+m∠PCA=RHS+m∠PCA$

$m∠A+m∠B=m∠PCA$

RearrangeEqn

Rearrange equation

$m∠PCA=m∠A+m∠B$

Consider $△ABC,$ where $D$ and $E$ are the midpoints of $AB$ and $AC,$ respectively. Let $∠PCA$ be one exterior angle of $△ABC.$

Now, $△ABC$ can be rotated $180_{∘}$ about $D.$ Since a rotation is a rigid motion, the image of $△ABC$ after the rotation is congruent to $△ABC.$ Corresponding parts of congruent figures are congruent, so the measures of the angles and the lengths of the sides remain unchanged.
Since a $180_{∘}-$rotation is equivalent to a reflection, $C_{′}A$ is parallel to $BC$ and $C_{′}B$ is parallel to $AC.$ Therefore, $C_{′}ACB$ is a parallelogram and $∠C_{′}AC$ is congruent to $∠CBC_{′}.$ Now the parallelogram $C_{′}ACB$ will be rotated $180_{∘}$ about $E.$

By the Parallelogram Opposite Angles Theorem, $∠PCA$ is congruent to $∠AB_{′′}P.$ Congruent angles have the same measure by the definition.

$∠PCAm∠PCA ≅∠AB_{′′}P⇕=m∠AB_{′′}P $

Since $m∠AB_{′′}P$ is equal to the sum of $m∠A$ and $m∠B$ and because of the Transitive Property of Equality, $m∠PCA$ is equal to the sum of $m∠A$ and $m∠B.$
${m∠PCA=m∠AB_{′′}Pm∠AB_{′′}P=m∠A+m∠B ⇓m∠PCA=m∠A+m∠B $

This completes the proof.
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 $90_{∘},$ the sofa is inclined backwards. Dylan is a bit busy with handling the wood. Help him find the measure of the exterior angle.

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Use the Triangle Exterior Angle Theorem.

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=35_{∘}+40_{∘}⇓x=75_{∘} $

Since the measure of the exterior angle is less than $90_{∘},$ the people sitting on this sofa can lean back and feel comfortable. Thanks for helping Dylan.

Given that $△ABC$ is a right triangle, reflect it across $AB.$

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

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.

If two sides of a triangle are congruent, then the angles opposite them are congruent.
The Isosceles Triangle Theorem is also known as the **Base Angles Theorem**.
### Proof

Geometric Approach

Therefore, $△BAP$ and $△CAP$ have two pairs of corresponding congruent sides and one pair of congruent included angles. By the Side-Angle-Side Congruence Theorem, $△BAP$ and $△CAP$ are congruent triangles.
### Proof

Using Transformations

The table shows that the images of the vertices of $△CAP$ are the vertices of $△BAP.$ It can be concluded that $△BAP$ is the image of $△CAP$ after a reflection across $AP.$ Since a reflection is a rigid motion, this proves that the triangles are congruent.

Based on this diagram, the following relation holds true.

$AB≅AC$ $⇒$ $∠B≅∠C$

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

In this triangle, let $P$ be the point of intersection of $BC$ and the angle bisector of $∠A.$ From the diagram, the following facts about $△BAP$ and $△CAP$ can be observed.

Statement | Reason |
---|---|

$∠BAP≅∠CAP$ | Definition of an angle bisector |

$BA≅CA$ | Given |

$AP≅AP$ | Reflexive Property of Congruence |

$△BAP≅△CAP $

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.
Consider an isosceles triangle $△ABC.$

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

Since $BP$ and $PC$ 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 $AP.$ Also, because $A$ lies on $AP,$ a reflection across $AP$ maps $A$ onto itself. The same is true for $P.$

Reflection Across $AP$ | |
---|---|

Preimage | Image |

$C$ | $B$ |

$A$ | $A$ |

$P$ | $P$ |

Corresponding parts of congruent figures are congruent, so $∠B$ and $∠C$ are congruent.

$∠B≅∠C $

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.

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.
### Proof

Using Coordinates ### $BC∥DE$

The $y-$coordinate of both $D(2b ,2c )$ and $E(2a+b ,2c )$ is $2c .$ Therefore, $DE$ is also a horizontal segment. Since all horizontal segments are parallel, it can be said that $BC$ and $DE$ are parallel.
### $DE=21 BC$

Since $21 a$ is half of $a,$ it can be stated that the midsegment $DE$ is half the length of $BC.$
### Proof

Using Transformations

If $DE$ is a midsegment of $△ABC,$ then the following statement holds true.

$DE∥BC$ and $DE=21 BC$

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

Since $B$ lies on the origin, its coordinates are $(0,0).$ Point $C$ is on the $x-$axis, meaning its $y-$coordinate is $0.$ The remaining coordinates are unknown and can be named $a,$ $b,$ and $c.$$B(0,0)C(a,0)A(b,c) $

If $DE$ is the midsegment from $BA$ to $CA,$ then by the definition of a midpoint, $D$ and $E$ are the midpoints of $BA$ and $CA,$ respectively.
To prove this theorem, it must be proven that $DE$ is parallel to $BC$ and that $DE$ is half of $BC.$

If the slopes of these two segments are equal, then they are parallel. The $y-$coordinate of both $B(0,0)$ and $C(a,0)$ is $0.$ Therefore, $BC$ is a horizontal segment. Next, the coordinates of $D$ and $E$ will be found using the Midpoint Formula.

$M(2x_{1}+x_{2} ,2y_{1}+y_{2} )$ | |||
---|---|---|---|

Segment | Endpoints | Substitute | Simplify |

$BA$ | $B(0,0)$ and $A(b,c)$ | $D(20+b ,20+c )$ | $D(2b ,2c )$ |

$CA$ | $C(a,0)$ and $A(b,c)$ | $E(2a+b ,20+c )$ | $E(2a+b ,2c )$ |

$BC∥DE✓ $

Since both $BC$ and $DE$ are horizontal, their lengths are given by the difference of the $x-$coordinates of their endpoints.

Segment | Endpoints | Length | Simplify |
---|---|---|---|

$BC$ | $B(0,0)$ and $C(a,0)$ | $BC=a−0$ | $BC=a$ |

$DE$ | $D(2b ,2c )$ and $E(2a+b ,2c )$ | $DE=2a+b −2b $ | $DE=21 a$ |

$DE=21 BC✓ $

Therefore, a midsegment of two sides of a triangle is parallel to the third side of the triangle and half its length.
This proof will be developed based on the given diagram, but it is valid for any triangle. ### $BC∥DE$

This part can be proven by using rigid motions. First, translate $△ADE$ along $DB$ so that $D$ is mapped onto $B.$ Since $D$ is the midpoint of $AB,$ $A$ is mapped onto $D.$
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To prove this theorem, it must be proven that $DE$ is parallel to $BC$ and that $DE$ is equal to half of $BC.$ Each statement will be proven one at a time.

Next, it must be proven that the image of $E$ — which is marked as $E_{′}$ — lies on $BC.$ This proof will be done using indirect reasoning. In this method, it is temporarily assumed that the negation of the statement is true.

1

Assume That $E_{′}$ Does Not Lie on $BC$

Assume that $E_{′},$ the image of $E$ after the translation along $DB,$ does not lie on $BC.$ This means that $E_{′}$ lies either *above* or *below* $BC.$ The proof will be developed for only one case but is valid for both.

Based on the assumption, let $F$ denote the point of intersection of $BE_{′}$ and $EC.$

Next, it will be proven that $△ADE$ is congruent to $△EE_{′}F.$

2

Show That $△ADE≅△EE_{′}F$

It is given that $AD=DB$ because $D$ is the midpoint of $AB.$ Additionally, since $△ADE$ is translated along $DB,$ it can be concluded that $DB=EE$