Here are a few recommended readings before getting started with this lesson.
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.
The opposite sides of a parallelogram are congruent.
In respects to the characteristics of the diagram, the following statement holds true.
$PQ ≅SRandQR ≅PS$
This theorem can also be proven by using congruent triangles. Consider the parallelogram PQRS and its diagonal PR.
It can be noted that two triangles are formed with PR as a common side.$PQ ≅SRandQR ≅PS$
Furthermore, it can be stated whether a quadrilateral is a parallelogram just by checking if its opposite sides are congruent.
If the opposite sides of a quadrilateral are congruent, then the polygon is a parallelogram.
Following the above diagram, the statement below holds true.
If PQ≅SR and QR≅PS, then PQRS is a parallelogram.
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.
Therefore, by the Side-Side-Side Congruence Theorem, △PQR and △RSP are congruent triangles.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.
This proves the theorem.
If PQ≅SR and QR≅PS, then PQRS is a parallelogram.
In a parallelogram, the opposite angles are congruent.
For the parallelogram PQRS, the following statement holds true.
$∠Q≅∠Sand∠P≅∠R$
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 ∠QPR≅∠SRP and ∠QRP≅∠SPR. 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.By drawing the diagonal QS and using a similar procedure, it can be shown that ∠P and ∠R are also congruent angles.
$∠Q≅∠Sand∠P≅∠R$
Furthermore, it can be determined whether a quadrilateral is a parallelogram just by looking at its opposite angles.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Based on the above diagram, the following statement holds true.
If ∠A≅∠C and ∠B≅∠D, then ABCD is a parallelogram.
Assume that ABCD is a quadrilateral with opposite congruent angles. It should be noted that congruent angles have the same measure. Then, let $x_{∘}$ be the measure of ∠A and ∠C, and $y_{∘}$ be the measure of ∠B and ∠D.
By the Polygon Interior Angles Theorem, the sum of the interior angles of a quadrilateral is $360_{∘}.$ With this information, a relation can be found between the consecutive interior angles of ABCD.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.
Find the values of a, b, and x to help Vincenzo be carefree for the match!First, for simplicity, the value of x will be found. After that, the values of a and b will be calculated.
$(II):$ LHS+10=RHS+10
$(I):$ a=3b
$(I):$ Multiply
$(I):$ LHS−10=RHS−10
$(I):$ LHS−10b=RHS−10b
$(I):$ $LHS/5=RHS/5$
$(II):$ b=10
$(II):$ Multiply
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.
In a parallelogram, the diagonals bisect each other.
If PQRS is a parallelogram, then the following statement holds true.
$PM≅RMandQM ≅SM$
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 ∠QPR≅∠SRP and that ∠PQS≅∠RSQ. Furthermore, by the Parallelogram Opposite Sides Theorem it can be said that PQ≅SR.
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.$PM≅RMandQM ≅SM$
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.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Based on the diagram above, the following relation holds true.
If AC and BD bisect each other, then ABCD is a parallelogram.
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.
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.
Applying a similar reasoning, it can be concluded that △AED and △CEB are congruent triangles. Consequently, AD and BC are also congruent.
Finally, since both pairs of opposite sides of quadrilateral ABCD are congruent, the Converse Parallelogram Opposite Sides Theorem states that ABCD is a parallelogram.
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.
Find the values of x and y and help Vincenzo finish his homework!According to the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each other.
With this information, two equations can be written.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.
A parallelogram is a rectangle if and only if its diagonals are congruent.
Based on the diagram, the following relation holds true.
PQRS is a rectangle ⇔ PR≅QS
Two proofs will be provided for this theorem. Each proof will consist of two parts.
This proof will use similar triangles to prove the theorem.
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.
Since the angles of a rectangle are right angles, by the definition of congruent angles, ∠RSP≅∠QPS. Consequently, △RSP and △QPS have two pairs of congruent sides and congruent included angles.Consider the parallelogram PQRS and its diagonals PR and QS such that PR≅QS.
By the Parallelogram Opposite Sides Theorem, PQ≅SR. Additionally, by the Reflexive Property of Congruence, PS is congruent to itself.
The sides of △QPS are congruent to the sides of △RSP.This proof will use transformations to prove the theorem.
Consider the rectangle PQRS and its diagonals PR and QS. Let M be the point of intersection of the diagonals.
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.
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 $AB.$ Similarly, P is the image of S after the same reflection.Reflection Across $AB$ | |
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Preimage | Image |
R | Q |
S | P |
M | M |
Consider the parallelogram PQRS and its diagonals PR and QS such that PR≅QS. By the Parallelogram Diagonals Theorem, the diagonals of a rectangle bisect each other at M.
By the Parallelogram Opposite Sides Theorem, PQ≅SR and QR≅PS.
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.
As shown before, Q, P, and M are the respective images of R, S, and M after a reflection across $AB.$ Therefore, since △QPM is the image of △RSM after a reflection across $AB,$ the triangles are congruent.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.
Find the lengths of its diagonals to help Zosia understand the stage that she will see her favorite artist sing on!In a rectangle, the diagonals are congruent.
As with rectangles, it can also be determined whether a parallelogram is a rhombus just by looking at its diagonals.
A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Based on the diagram, the following relation holds true.
Parallelogram ABCD is a rhombus ⇔
This proof will be written in two parts.
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.
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.Parallelogram ABCD is a rhombus ⇒ $AC⊥BD$
Conversely, let ABCD be a parallelogram whose diagonals are perpendicular.
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.
Since AC and BD are perpendicular, ∠APB and ∠CPB measure $90_{∘}$ 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.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.
This means that if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
$AC⊥BD$ ⇒ parallelogram ABCD is 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.
Help Zosia draw the perfect design by finding the length of AB!If the diagonals of a parallelogram are perpendicular, then the quadrilateral is a rhombus.
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.
Therefore, by the Rectangle Diagonals Theorem and the Rhombus Diagonals Theorem, the diagonals of a square are congruent and perpendicular.