PROPERTIES OF PARALLELOGRAM
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. When we mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel.
For example, in the diagram shown below,
AB || CD
AD || BC
Properties of Parallelogram
Property 1 :
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
It has been illustrated in the diagram shown below.
In the diagram above,
AB ≅ DC
AD ≅ BC
Property 2 :
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
It has been illustrated in the diagram shown below.
In the diagram above,
m∠A ≅ m∠C
m∠B ≅ m∠D
Property 3 :
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
It has been illustrated in the diagram shown below.
In the diagram above,
m∠A + m∠B = 180°
m∠B + m∠C = 180°
m∠C + m∠D = 180°
m∠A + m∠D = 180°
Property 4 :
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
It has been illustrated in the diagram shown below.
In the diagram above,
AM ≅ CM
BM ≅ DM
Practice Problems
Problem 1 :
Find the lengths of SR and SK in the parallelogram shown below. Explain your reasoning.
Solution :
Finding the length of SR :
By Theorem, opposite sides of a parallelogram are congruent.
So, we have
SR = PQ
From the diagram, PQ = 5.
SR = 5
Finding the length of SK :
By Theorem, diagonals of a parallelogram bisect each other.
So, we have
SK = QK
From the diagram, QK = 3.
SK = 3
Problem 2 :
Find the measures of ∠C and ∠B in the parallelogram ABCD.
Solution :
Finding the measure of ∠C :
By Theorem, opposite angles of a parallelogram are congruent.
So, we have
m∠C = m∠A
From the diagram, m∠A = 70°.
m∠C = 70°
Finding the measure of ∠B :
By Theorem, consecutive angles of a parallelogram are supplementary.
So, we have
m∠A + m∠B = 180°
From the diagram, m∠A = 70°.
70° + m∠B = 180°
Subtract 70° from both sides.
m∠B = 110°
Problem 3 :
Find the value of x in the parallelogram ABCD shown below.
Solution :
By Theorem, consecutive angles of a parallelogram are supplementary.
So, we have
m∠D + m∠C = 180°
Substitute m∠D = 3x° and m∠C = 120°.
3x° + 120° = 180°
Subtract 120° from both sides.
3x° = 60°
3x = 60
Divide both sides by 3.
x = 20
Problem 4 :
In the diagram shown below, ABCD and AEFG are parallelograms. Prove that m∠1 ≅ m∠3.
Solution :
Plan : Show that both angles are congruent to m∠2. Then use the Transitive Property of Congruence.
|
Statements ABCD is a parallelogram AEFG is a parallelogram aaaaa m∠1 ≅ m∠2 aaa aaaaa m∠2 ≅ m∠3 aaa m∠1 ≅ m∠2 |
Reasons aaaaaaaaa Given aaaaaaa aaaaaaaaaaaaaaaaaaaaaaa Opposite angles of a parallelogram are congruent. Transitive property of congruence. |
Problem 5 :
In the parallelogram ABCD shown below, prove that WX ≅ YZ and WZ ≅ XY.
|
Statements aaaaaa WXYZ is a aaaaa aaaa parallelogram aaaa aaaaaa Draw XZ aaaaaa aaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaa WX || YZ, WZ || XY aa m∠WXZ ≅ m∠YZX a aa m∠WZX ≅ m∠YXZ aaaaaa ZX ≅ ZX aaaaa aaaaaaaaaaaaaaaaaaaaa ΔWZX ≅ ΔYXZ WX ≅ YZ, WZ ≅ XY |
Reasons aaaaaaaaaa Given aaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaa aaThrough any two points there exists exactly one a aaaaaaaaa line. Definition of parallelogram Alternate Interior Angles Theorem Reflexive Property of Congruence ASA Congruence Postulate Corresponding parts of congruent triangles are congruent |
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and Venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
©All rights reserved. onlinemath4all.com
Recent Articles
-
Exponential vs Linear Growth Worksheet
May 23, 22 01:59 AM
Exponential vs Linear Growth Worksheet
-
Linear vs Exponential Growth
May 23, 22 01:59 AM
Linear vs Exponential Growth - Concept - Examples
-
SAT Math Questions on Exponential vs Linear Growth
May 23, 22 01:34 AM
SAT Math Questions on Exponential vs Linear Growth