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Congruence, Proof, and Constructions

Theorems About Parallelograms

It may not be understood at first glance, but all squares, rectangles, and rhombi are parallelograms. Therefore, all the properties of parallelograms apply to these quadrilaterals as well! In this lesson, theorems about parallelograms will be discussed to understand this concept better.

Catch-Up and Review

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


Explore

Investigating Properties of Parallelograms

In the applet below, a parallelogram, a rectangle, a rhombus, and a square can be selected and rotated clockwise about the point of intersection of its diagonals. What can be noted when these polygons are rotated
quadrilaterals
Note that all the quadrilaterals in the applet are parallelograms. Additionally, when rotated each quadrilateral is mapped onto itself. Considering this information, what conclusions can be made about the opposite sides of a parallelogram?

Discussion

Properties of a Parallelogram's Opposite Sides

A conclusion that can be made from the previous exploration is that the opposite sides of a parallelogram are congruent. This is explained in detail in the following theorem.

Rule

Parallelogram Opposite Sides Theorem

The opposite sides of a parallelogram are congruent.

parallelogram

In respects to the characteristics of the diagram, the following statement holds true.

Proof

Using Congruent Triangles

This theorem can also be proven by using congruent triangles. Consider the parallelogram PQRS and its diagonal PR.

parallelogram and one of its diagonals
It can be noted that two triangles are formed with PR as a common side.
By the definition of a parallelogram, PQ and SR are parallel. Therefore, by the Alternate Interior Angles Theorem, it can be stated that QPRSRP and that QRPSPR. Furthermore, by the Reflexive Property of Congruence, PR is congruent to itself.
parallelogram and one of its diagonals and the two pair of congruent angles
Consequently, PQR and RSP have two pairs of congruent angles and an included congruent side.
Therefore, by the Angle-Side-Angle Congruence Theorem, PQR and RSP are congruent triangles.
Since corresponding parts of congruent figures are congruent, PS is congruent to QR and PQ is congruent to RS.

Furthermore, it can be stated whether a quadrilateral is a parallelogram just by checking if its opposite sides are congruent.

Rule

Converse Parallelogram Opposite Sides Theorem

If the opposite sides of a quadrilateral are congruent, then the polygon is a parallelogram.

parallelogram

Following the above diagram, the statement below holds true.

If PQSR and QRPS, then PQRS is a parallelogram.

Proof

This theorem can be proven by using congruent triangles. Consider the quadrilateral PQRS, whose opposite sides are congruent, and its diagonal PR. By the Reflexive Property of Congruence, this diagonal is congruent to itself.

parallelogram and one of its diagonals
Therefore, by the Side-Side-Side Congruence Theorem, PQR and RSP are congruent triangles.
Since corresponding parts of congruent figures are congruent, corresponding angles of PQR and RSP are congruent.
parallelogram and one of its diagonals

Finally, by the Converse of the Alternate Interior Angles Theorem, PQ is parallel to RS and QR is parallel to SP. Therefore, by the definition of a parallelogram, PQRS is a parallelogram.

parallelogram

This proves the theorem.

If PQSR and QRPS, then PQRS is a parallelogram.

Another conclusion that can be made from the exploration is that the opposite angles of a parallelogram are congruent. This is explained in detail in the following theorem.

Rule

Parallelogram Opposite Angles Theorem

In a parallelogram, the opposite angles are congruent.

For the parallelogram PQRS, the following statement holds true.

Proof

This theorem can be proved by using congruent triangles. Consider the parallelogram PQRS and its diagonal PR.

Opposite sides of a parallelogram are parallel. Therefore, by the Alternate Interior Angles Theorem it can be stated that QPRSRP and QRPSPR. Furthermore, by the Reflexive Property of Congruence, PR is congruent to itself.

Two angles of PQR and their included side are congruent to two angles of RSP and their included side. By the Angle-Side-Angle Congruence Theorem, PQR and RSP are congruent triangles.
Since corresponding parts of congruent figures are congruent, Q and S are congruent angles.

By drawing the diagonal QS and using a similar procedure, it can be shown that P and R are also congruent angles.

Furthermore, it can be determined whether a quadrilateral is a parallelogram just by looking at its opposite angles.

Rule

Converse Parallelogram Opposite Angles Theorem

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A parallelogram with congruent opposite angles

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

If AC and BD, then ABCD is a parallelogram.

Proof

Assume that ABCD is a quadrilateral with opposite congruent angles. It should be noted that congruent angles have the same measure. Then, let be the measure of A and C, and be the measure of B and D.

By the Polygon Interior Angles Theorem, the sum of the interior angles of a quadrilateral is With this information, a relation can be found between the consecutive interior angles of ABCD.
x+y+x+y=360
Simplify
2x+2y=360
2(x+y)=360
x+y=180
Since x+y=180, the consecutive interior angles of ABCD are supplementary. Therefore, by the Converse Consecutive Interior Angles Theorem, it can be concluded that opposite sides of ABCD are parallel.
A polygon ABCD with one movable vertex
Consequently, by the definition of a parallelogram, ABCD is a parallelogram.

Completed Proof

Proof:
Proof of the theorem

Example

Solving Problems Using Properties of a Parallelogram

To be able to be carefree and enjoy a soccer match over the weekend, Vincenzo wants to complete his Geometry homework immediately after school. He is given a diagram showing a parallelogram, and asked to find the values of a, b, and x.

parallelogram
Find the values of a, b, and x to help Vincenzo be carefree for the match!

Hint

In a parallelogram, opposite sides are congruent and opposite angles are congruent.

Solution

First, for simplicity, the value of x will be found. After that, the values of a and b will be calculated.

Value of x

According to the Parallelogram Opposite Sides Theorem, the opposite sides of a parallelogram are congruent. Therefore, the opposite sides have the same length. With this information, an equation in terms of x can be expressed.
This equation can now be solved for x.
2x+5=5x10
Solve for x
-3x+5=-10
-3x=-15
x=5

Values of a and b

According to the Parallelogram Opposite Angles Theorem, the opposite angles of a parallelogram are congruent. That means the opposite angles have the same measure. Knowing this, a system of equations can be expressed.
This system will be solved by using the Substitution Method. For simplicity, the degree symbol will be removed.
Solve by substitution

Explore

Investigating the Diagonals of a Parallelogram

In the following applet, a parallelogram is shown. By dragging one of its vertices, different types of parallelograms such as squares, rectangles, and rhombi can be formed. By using the measuring tool provided, investigate what relationships exist between the diagonals of each parallelogram.
parallelogram
After investigating each parallelogram type, what relationships between the diagonals of the parallelograms were discovered?

Discussion

Properties of a Parallelogram's Diagonals

A conclusion that can be made from the previous exploration is that the diagonals of a parallelogram intersect at their midpoint. This is explained in detail in the following theorem.

Rule

Parallelogram Diagonals Theorem

In a parallelogram, the diagonals bisect each other.

If PQRS is a parallelogram, then the following statement holds true.


Proof

Using Congruent Triangles

This theorem can be proven by using congruent triangles. Consider the parallelogram PQRS and its diagonals PR and QS. Let M be the point intersection of the diagonals.

Since PQ and SR are parallel, by the Alternate Interior Angles Theorem it can be stated that QPRSRP and that PQSRSQ. Furthermore, by the Parallelogram Opposite Sides Theorem it can be said that PQSR.

Here, two angles of PMQ and their included side are congruent to two angles of RMS and their included side. Therefore, by the Angle-Side-Angle Congruence Theorem PMQ and RMS are congruent triangles.
Since corresponding parts of congruent triangles are congruent, PM is congruent to RM and QM is congruent to SM.

By the definition of a segment bisector, both segments PR and QS are bisected at point M. Therefore, it has been proven that the diagonals of a parallelogram bisect each other.

Also, a quadrilateral can be identified as a parallelogram just by looking at its diagonals.

Rule

Converse Parallelogram Diagonals Theorem

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Quadrilateral with diagonals that bisect each other

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

If AC and BD bisect each other, then ABCD is a parallelogram.

Proof

Let E be point of intersection of the diagonals of a quadrilateral. Since the diagonals bisect each other, E is the midpoint of each diagonal.

Quadrilateral with diagonals that bisect each other

Because AEB and CED are vertical angles, they are congruent by the Vertical Angles Theorem. Therefore, by the Side-Angle-Side Congruence Theorem, AEB and CED are congruent triangles. Since corresponding parts of congruent figures are congruent, AB and CD are congruent.

Quadrilateral with diagonals that bisect each other

Applying a similar reasoning, it can be concluded that AED and CEB are congruent triangles. Consequently, AD and BC are also congruent.

Quadrilateral with diagonals that bisect each other

Finally, since both pairs of opposite sides of quadrilateral ABCD are congruent, the Converse Parallelogram Opposite Sides Theorem states that ABCD is a parallelogram.

Quadrilateral with diagonals that bisect each other

Example

Solving Problems Using Properties of a Parallelogram's Diagonals

Vincenzo has one last exercise to finish before going to a soccer match. He has been given a diagram showing a parallelogram. He is asked to find the value of x and y.

parallelogram
Find the values of x and y and help Vincenzo finish his homework!

Hint

The diagonals of a parallelogram bisect each other.

Solution

According to the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each other.

parallelogram
With this information, two equations can be written.
These can be solved one at a time. Equation (I) will be solved first.
9=2x+1
Solve for x
8=2x
4=x
x=4
The value of x is 4. Finally, Equation (II) will be solved.
y+4=3y
Solve for y
4=2y
2=y
y=2
The value of y is 2.

Discussion

Investigating Rotations of Parallelograms

Consider a rigid motion that rotates a parallelogram about the point of intersection of its diagonals.
parallelogram
Since a rotation is a rigid motion, the preimage and the image are congruent figures. Furthermore, because corresponding parts of congruent figures are congruent, the three statements below hold true.
  • Opposite sides of a parallelogram are congruent.
  • Opposite angles of a parallelogram are congruent.
  • The diagonals of a parallelogram bisect each other.
Note that the Parallelogram Opposite Sides Theorem, the Parallelogram Opposite Angles Theorem, and the Parallelogram Diagonals Theorem have been proved using a rotation about the point of intersection of the diagonals.

Discussion

Diagonals of a Rectangle

It can be determined whether a parallelogram is a rectangle just by looking at its diagonals. Furthermore, if a parallelogram is a rectangle, a statement about its diagonals can be made.

Rule

Rectangle Diagonals Theorem

A parallelogram is a rectangle if and only if its diagonals are congruent.

rectangle with its diagonals marked

Based on the diagram, the following relation holds true.

PQRS is a rectangle PRQS

Two proofs will be provided for this theorem. Each proof will consist of two parts.

  • Part I: If PQRS is a rectangle, then PRQS.
  • Part II: If PRQS, then PQRS is a rectangle.

Proof

Using Similar Triangles

This proof will use similar triangles to prove the theorem.

Part I: PQRS Is a Rectangle PRQS

Suppose PQRS is a rectangle and PR and QS are its diagonals. By the Parallelogram Opposite Sides Theorem, the opposite sides of a parallelogram are congruent. Therefore, RS and QP are congruent. Additionally, by the Reflexive Property of Congruence, SP, or PS, is congruent to itself.

rectangle with its diagonals marked
Since the angles of a rectangle are right angles, by the definition of congruent angles, RSPQPS. Consequently, RSP and QPS have two pairs of congruent sides and congruent included angles.
Therefore, by the Side-Angle-Side Congruence Theorem, the triangles are congruent.
Because corresponding parts of congruent triangles are congruent, PR and QS, which are the diagonals of PQRS, are congruent.

Part II: PRQSPQRS Is a Rectangle

Consider the parallelogram PQRS and its diagonals PR and QS such that PRQS.

rectangle with its diagonals marked

By the Parallelogram Opposite Sides Theorem, PQSR. Additionally, by the Reflexive Property of Congruence, PS is congruent to itself.

rectangle with its diagonals marked
The sides of QPS are congruent to the sides of RSP.
Therefore, by the Side-Side-Side Congruence Theorem, QPSRSP. Moreover, since corresponding parts of congruent triangles are congruent, QPS is congruent to RSP.
Note that QPS and RSP are consecutive angles. By the Parallelogram Consecutive Angles Theorem, these angles are supplementary. With this information, it can be concluded that both QPS and RSP are right angles.
Additionally, by the Parallelogram Opposite Angles Theorem, QPSSRQ and RSPPQR. Because all of the angles are right angles, PQRS is a rectangle.

Proof

Using Transformations

This proof will use transformations to prove the theorem.

Part I: PQRS Is a Rectangle PRQS

Consider the rectangle PQRS and its diagonals PR and QS. Let M be the point of intersection of the diagonals.

rectangle with its diagonals marked

Let A and B be the midpoints of PS and RQ. Then, a line through M and the midpoints A and B can be drawn.

rectangle with its diagonals marked
Note that QA, AR, PB, and BS are congruent segments. Because congruent segments have the same length, the distance between Q and A equals the distance between R and A. Therefore, Q is the image of R after a reflection across Similarly, P is the image of S after the same reflection.
rectangle with its diagonals marked
Since M lies on a reflection across maps M onto itself.
Reflection Across
Preimage Image
R Q
S P
M M
The table shows that the images of the vertices of RSM are the vertices of QPM. Therefore, QPM is the image of RSM after a reflection across Since a reflection is a rigid motion, this proves that the triangles are congruent.
rectangle with its diagonals marked
Because corresponding parts of congruent figures are congruent, QMRM and PMSM. Additionally, by the Parallelogram Diagonals Theorem, the diagonals of the rectangle bisect each other. Therefore, all four segments are congruent.
rectangle with its diagonals marked
Each diagonal of the parallelogram consists of the same two congruent segments. By the Segment Addition Postulate, the diagonals are congruent.

Part II: PRQSPQRS Is a Rectangle

Consider the parallelogram PQRS and its diagonals PR and QS such that PRQS. By the Parallelogram Diagonals Theorem, the diagonals of a rectangle bisect each other at M.

rectangle with its diagonals marked

By the Parallelogram Opposite Sides Theorem, PQSR and QRPS.

rectangle with its diagonals marked

Let A and B be the midpoints of PS and RQ. Then, a line through M and the midpoints A and B can be drawn.

rectangle with its diagonals marked
As shown before, Q, P, and M are the respective images of R, S, and M after a reflection across Therefore, since QPM is the image of RSM after a reflection across the triangles are congruent.
rectangle with its diagonals marked
Let C and D be the midpoints of of PQ and RS. By following the same reasoning, PMS is the image of QMR after a reflection across Therefore, the triangles are congruent.
rectangle with its diagonals marked
The parallelogram consists of four triangles in which the opposite triangles are congruent. Therefore, the corresponding angles of these triangles are congruent. Additionally, because all triangles are isosceles, the angles opposite congruent sides are congruent as well.
rectangle with its diagonals marked
Each angle of the parallelogram is the sum of the same two congruent angles. Therefore, all angles of the parallelogram are congruent.
Moreover, by the Parallelogram Consecutive Angles Theorem, P and S are supplementary. With this information, it can be concluded that both angles are right triangles.
Because all of the angles are congruent, the angles of the parallelogram are right triangles. Therefore, PQRS is a rectangle.

Example

Solving Problems Using the Diagonals of a Rectangle

Zosia arrives early to a Harry Styles concert! She notices something about the stage, so she uses a napkin as paper and draws a diagram. The stage is a rectangle that she labels as ABCD.

rectangle
Find the lengths of its diagonals to help Zosia understand the stage that she will see her favorite artist sing on!

Hint

In a rectangle, the diagonals are congruent.

Solution

By the Rectangle Diagonals Theorem, the diagonals of a rectangle are congruent. This means that AC and BD have the same length. With this information, an equation in terms of x can be written.
The above equation will be solved for x.
4x+3=452x
Solve for x
6x+3=45
6x=42
x=7
The value of x was found. Next, this value can be substituted in any of the expressions for the length of a diagonal. In this case, x=7 will be arbitrarily substituted in the expression for AC.
AC=4x+3
AC=4(7)+3
Evaluate right-hand side
AC=28+3
AC=31
It was found that the length of AC is 31. Since the diagonals of a rectangle are congruent, the length of BD is also 31.

Discussion

Diagonals of a Rhombus

As with rectangles, it can also be determined whether a parallelogram is a rhombus just by looking at its diagonals.

Rule

Rhombus Diagonals Theorem

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Parallelogram with diagonals drawn

Based on the diagram, the following relation holds true.

Parallelogram ABCD is a rhombus

Proof

This proof will be written in two parts.

If a Parallelogram Is a Rhombus, Then Its Diagonals Are Perpendicular

A rhombus is a parallelogram with four congruent sides. By the Parallelogram Diagonals Theorem, it can be said that its diagonals bisect each other. Let Let ABCD be a rhombus with P at the midpoint of both diagonals.

Rhombus with diagonals and midpoint that bisect diagonals
Note that AB is congruent to AD and DP is congruent to BP. Additionally, by the Reflexive Property of Congruence, AP is congruent to itself. Therefore, by the Side-Side-Side Congruence Theorem, APB is congruent to APD.
Because corresponding parts of congruent triangles are congruent, APB and APD are congruent angles. Furthermore, these angles form a linear pair, which means they are supplementary. With this information, it can be concluded that both APB and APD are right angles.
This implies that AC is perpendicular to BD. Therefore, the diagonals of a rhombus are perpendicular.

Parallelogram ABCD is a rhombus

If Its Diagonals Are Perpendicular, Then a Parallelogram is a Rhombus

Conversely, let ABCD be a parallelogram whose diagonals are perpendicular.

Parallelogram with perpendicular diagonals

By the Parallelogram Diagonals Theorem, the diagonals of the parallelogram bisect each other. If P is the midpoint of both diagonals, then AP and CP are congruent.

Parallelogram with perpendicular diagonals and mindpoint of diagonals
Since AC and BD are perpendicular, APB and CPB measure and thus are congruent angles. By the Reflexive Property of Congruence, BP is congruent to itself. This means that two sides and their included angle are congruent. By the Side-Angle-Side Congruence Theorem, APB and CPB are congruent triangles.
Because corresponding parts of congruent figures are congruent, it can be said that AB is congruent to CB.
Parallelogram with perpendicular diagonals and mindpoint of diagonals

Furthermore, by the Parallelogram Opposite Sides Theorem, AB is congruent to DC and AD is congruent to BC. By the Transitive Property of Congruence, it follows that all sides of the parallelogram are congruent.

Parallelogram with perpendicular diagonals and mindpoint of diagonals

This means that if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

parallelogram ABCD is a rhombus

Example

Solving Problems Using the Diagonals of a Rhombus

Zosia is now listening to Dua Lipa at home. Staring at some of her album covers, Zosia decides to design a parallelogram as the background art for Dua's next cover! She has made a parallelogram ABCD in which the diagonals are perpendicular. To make a unique design, she wants to be sure of the length of AB.

rhombus
Help Zosia draw the perfect design by finding the length of AB!

Hint

If the diagonals of a parallelogram are perpendicular, then the quadrilateral is a rhombus.

Solution

By the Rhombus Diagonal Theorem, the diagonals of a parallelogram are perpendicular if and only if the quadrilateral is a rhombus. Therefore, since BD and AC are perpendicular, ABCD is a rhombus. This means that BC and CD have the same length. With this information, an equation in terms of x can be written.
The above equation will be now solved for x.
6x+3=8x19
-2x+3=-19
-2x=-22
x=11
The value of x was found. Since the given quadrilateral is a rhombus, all sides are congruent and therefore have the same length. This means that the length of AB is the same as the length of BC. To find it, x=11 will be substituted into the expression for BC.
BC=6x+3
BC=6(11)+3
Evaluate right-hand side
BC=66+3
BC=69
The length of BC is 69. As it has been previously said, all sides have the same length. Therefore, AB is also 69.

Closure

Additional Applications of a Parallelogram's Properties

By using the theorems seen in this lesson, other properties can be derived. One of them is the Parallelogram Consecutive Angles Theorem.

Parallelogram Consecutive Angles Theorem

The consecutive angles of a parallelogram are supplementary.

Furthermore, the theorems seen in this lesson can be applied to different parallelograms in different contexts. Consider a square. By definition, all its angles are right angles, and all its sides are congruent. Therefore, a square is both a rectangle and a rhombus.

square

Therefore, by the Rectangle Diagonals Theorem and the Rhombus Diagonals Theorem, the diagonals of a square are congruent and perpendicular.

square
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