Current Affairs 7th Class

INTEGERS   FUNDAMENTALS

• In lower classes, you would have read about counting number. 1, 2, 3,.........
• They are called natural numbers (N).

N= (1, 2, 3, 4,............)   Elementary Question - 1: Which is the smallest natural number? Ans.: 1  

• Representation of natural numbers on a number line. To represent natural numbers on a number line, we should draw a line and write the number at equal distances on it as shown below:

     

• Whole Number (W): The set of natural numbers together with zero is known as the set of whole numbers.

  Elementary Question - 2: Which is the smallest whole number? Ans.: 0 W= (0, 1, 2, 3,............)

• In set notation, set of whole numbers (W) = set of natural numbers (N) + zero; { } is used for set notation.
• Integers (Z): The set containing negatives of natural numbers along with whole numbers is called the set of integers.

That is, \[Z=\{........-4,\,\,-3,\,\,-2,\,\,-1\}\cup \{0,\text{ }1,\,\,2,\,\,3,\,\,4,\,\,5,.....\}\] Where \[\cup \] denote "union" or combination of these two sets.   Concept of Infinity We can go on adding more and more numbers to the right side of the number line (e.g., 100, 101, ......100000, .........1 crore, ............1000 crores............. m an unending manner upto plus infinity and similarly to the left side of the number line upto minus infinity.     \[\left( -\,\infty  \right)\]Minus infinity crore  Plus infinity \[\left( +\infty  \right)\] This very, very large unending number on the right side and left side of number line are called plus infinity \[\left( +\,\infty  \right)\]and minus infinity \[\left( -\,\infty  \right)\] respectively. \[\left( -\,\infty  \right)\] \[\left( +\,\infty  \right)\]   Note:

Usually, negative numbers are placed in brackets to avoid confusion arising due to two signs in evaluations simultaneously,

e.g., \[3+\left( -\,5 \right)=-2\] 2.0 is not included in either \[{{Z}^{+}}\]c or\[{{Z}^{-}}\]. Hence, it is non-negative integer   Common use of numbers (i) To represent quantities like profit, income, increase, rise, high, north, east, above, depositing, climbing and so on, positive numbers are used. (ii) To represent quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on, negative numbers are used. (iii) On a number line, when we

Add a positive integer, we move to the right.

Add a negative integer, we move to the left.

Subtract a positive integer, we move to the left.

Subtract a negative integer, we move to the right

Notation e means belongs to; a, b \[\in \] means the numbers a and b belong to \[\] (set of integers)

  Note: 

0 is neither positive nor negative.

The + sign is not written before a positive number

\[\frac{1}{2}\] and 0.3 are more...

RATIONAL NUMBERS   FUNDAMENTALS   

• Natural numbers (N): 1, 2, 3, 4, 5..... ..etc., are called natural numbers.
• Whole numbers (W): 0, 1, 2, 3, 4, etc.., are called whole numbers.
• Integers (Z): 1.......\[-4,-3,-2,-1,\,\,0,\,\,1,\,\,2,\,\,3,\,\,4\]........ etc.., are called integers. (denoted by I or Z) 1, 2, 3, 4, .. ...etc., are called positive integers denoted by \[{{Z}^{+}}\]or \[{{I}^{+}}\].

\[-1,\,\,-2,\,\,-3,\,\,-4,\].......etc.., are called negative integers denoted by \[{{Z}^{-}}\] or \[{{I}^{-}}\].   Note: 0 is neither positive nor negative numbers.

• Fractions: The numbers of the form \[\frac{x}{y}\], where \[x\] and \[y\]c are natural numbers, are known as fractions. e.g., \[\frac{2}{5},\,\,\frac{3}{1},\,\,\frac{1}{122},.....\]etc.

Elementary questions: Identify which of the following number is a whole number as well as a fraction? (a) \[\frac{3}{36}\]                                 (b) \[\frac{36}{3}\]                                 (c) \[\frac{20}{8}\]                                      (d) \[\frac{8}{20}\]   Ans. (b) \[\frac{36}{3}=12\] which can be expressed as a fraction \[\left( \frac{12}{1} \right)\] as well as a whole number (=12). Rational numbers (Q): A number of the form \[\frac{p}{q}(q\ne 0).\] where p and q are integers is called a rational number. e.g., \[\frac{-3}{6},-\frac{1}{12},\frac{10}{13},\frac{12}{17},\ldots \ldots ..\]etc. Note: 0 is rational number, since \[0=\frac{0}{1}.\]

• A rational number \[\frac{p}{q}\] is positive if p and q are either both positive or both negative.

e.g. \[\frac{6}{11},\frac{-8}{-16}\]

• A rational number \[\frac{p}{q}\] is negative if either of p and q is positive and the other term (q or p) is negative.

e.g., \[\frac{-4}{7},\frac{8}{-23}\]   Note: 0 is neither a positive nor a negative rational number.

• Representation of Rational numbers on a number line:

We can mark rational numbers on a number line just as we do for integer. The negative rational numbers are marked to the left of 0 and the positive rational numbers are marked to the right of 0. Thus, \[\frac{1}{6}\] and \[-\,\,\frac{1}{6}\] would be at an equal distance from 0 but on its either side of zero. Similarly, other rational numbers with different denominators can also be represented on the number line.

• In general, any rational number is either of the following two types.

(a) \[\frac{p}{q}\] where p < q                            (b) \[\frac{p}{q}\] where p > q e.g., \[\frac{1}{8},\frac{2}{9},\frac{16}{17}\] etc.                                   e.g.,\[\frac{8}{1},\frac{9}{2},\frac{17}{16}\]etc. Representation of \[\frac{p}{q}\] on the number line where p < q: The rational number \[\frac{4}{6}\](4<6) is represented on the number line as shown.   Representation of \[\frac{p}{q}\]on the number line where p > q: Consider the rational number \[\frac{13}{6}\] Let us convert the rational number \[\frac{13}{6}\] into a mixed fraction \[=2\frac{1}{6}\]and then mark it on the number line. i.e.

• Standard form of a rational number:

A rational number \[\frac{p}{q}\] is said to be in standard form if q is a positive integer and the integer p and q have no common factor other than 1.   Additive Inverse: \[\frac{-\,p}{q}\]is the additive inverse of \[\frac{p}{q}\] and \[\frac{p}{q}\] is the additive inverse of \[\frac{-p}{q}.\] more...

FRACTIONS AND DECIMALS   FUNDAMENTALS

• A fraction is a part of a whole.
• A number of the form\[\frac{p}{q}\], where p and q are whole numbers and \[q\ne 0\]is known as a fraction.
• In the fraction\[\frac{p}{q}\], p is called the numerator and q is called the denominator.

  Various Types of Fractions

Improper fraction: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction. If number is written as\[\frac{N}{D}\], then \[N~\underline{>}\,D\]where N = numerator D = Denominator e.g.,\[\frac{100}{100},\] \[\frac{51}{50},\] \[\frac{45}{2},.....\]etc.

Proper fraction: A fraction whose numerator is less than its denominator is called a proper fraction. e.g.,\[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{18},......\]etc.

Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc., is called a decimal fraction. e.g., \[\frac{2}{10},\,\,\frac{8}{100},\,\,\frac{26}{1000},\,\,\frac{1312}{1000},......\]etc.

Vulgar fraction: A fraction whose denominator is a whole number other than 10, 100, 1000, etc., is called a vulgar fraction. e.g., \[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{83}{103},\,\,......\]etc.

Simple fraction: A fraction in its lowest terms is known as a simple fraction. e.g., \[\frac{3}{8},\,\,\frac{6}{7},\,\,\frac{9}{16},\,\,\frac{8}{17},\,\,......\]etc.

Mixed fraction: A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.

e.g., \[5\frac{2}{3},\,\,6\frac{1}{6},\,\,1\frac{3}{4},\,\,107\frac{1}{2},\,\,.......\]etc.            \[5\frac{2}{3}\]can be written as \[5+\frac{2}{3}\]

Like fractions: Fractions having the same denominator but different numerators are called like fractions. e.g.\[\frac{6}{17},\,\,\frac{9}{17},\,\,\frac{11}{17},\,\,......\]etc.

Unlike fractions: Fractions having different denominators are called unlike fractions. e.,\[\frac{3}{4},\,\,\frac{6}{7},\,\,\frac{5}{6},\,\,\frac{4}{5},\,\,\frac{8}{13},\,\,.......\]etc.

• An important property: If the numerator and denominator of a fraction are both multiplied by the same non zero number, its value is not changed. Thus, \[\frac{2}{7}=\frac{2\times 2}{7\times 2}=\frac{2\times 5}{7\times 5}......\]etc.
• Equivalent fractions: A given fraction and the fraction obtained by multiplying (or dividing) its numerator and denominator by the same non — zero number, are called equivalent fractions. (See above property) e.g., Equivalent fractions of \[\frac{6}{12}\]are \[\frac{8}{16},\frac{1}{2},\frac{4}{8},\frac{5}{10}\]......etc.

 

• Method of changing unlike fraction to like fraction:

Step - 1: Find the L.C.M. of the denominators of all the given fractions. Step - 2: Change each of the given fractions into an equivalent fraction having denominator equal to the L.C.M. of the denominators of the given fraction. e.g., Convert the fractions \[\frac{1}{6},\,\,\frac{4}{9}\] and \[\frac{70}{12}\] into like fractions. L.C.M. of 6, 9 and \[12=3\times 2\times 3\times 2=36\] Now, \[\frac{1}{6}=\frac{1\times 6}{6\times 6}=\frac{6}{36};\frac{4}{9}=\frac{4\times 4}{9\times 4}=\frac{16}{36}\] and\[\frac{70}{12}=\frac{70\times 3}{12\times 3}=\frac{21}{36}\]. Clearly, \[\frac{6}{36},\frac{16}{36}\] and \[\frac{21}{36}\] are like fractions. Utility: This method is used for comparing fractions

• Irreducible fractions : A fraction \[\frac{a}{b}\] is said to be irreducible or in lowest terms,

If the H.C.F. of a and b is 1. They are also called simple fractions. If H.C.F. of a and b is not 1, then \[\frac{a}{b}\] is said to be reducible.

• Comparing fraction: Let \[\frac{a}{b}\] and \[\frac{c}{d}\] be two given fractions. Then,

(a) \[\frac{a}{b}\text{}\frac{c}{d}\text{ }\Leftrightarrow ad>bc~~\]                     (b) \[\frac{a}{b}\text{=}\frac{c}{d}\text{ }\Leftrightarrow ad=bc~~\]                       (c) \[\frac{a}{b}\text{}\frac{c}{d}\Leftrightarrow \text{ }ad<bc\] These are important results and should be committed to memory. (or understood as coming out of cross = multiplication).  

• Method of comparing more than two fractions;

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LINEAR EQUATION IN ONE VARIABLE   FUNDAMENTALS  

• Variable: A symbol which takes various values is known as a variable. Normally it is denoted by letters x, y etc.
• Constant: A symbol having a fixed numerical value is called a constant.

Symbols used to denote a constant are generally, 'c', 'k' etc...

• Coefficient: In the product of a variable and a constant, each is called the coefficient of the other. Sometimes, symbols like a, b, 1, m etc..., are used to denote the coefficients.
• Expression: An expression can be defined as a combination of constant, variables and coefficients by some or all of the four fundamental mathematical operations \[(+,\,\,-,\,\,\times \,\,\text{and}\,\,\div ).\]

e.g., 2x - 6; here, 2 is the coefficient of \['x';\,\,'x'\] is the variable and \[-6\]is the constant. Similarly in \[ay+b;\,\,a\] is the coefficient of \[y;\,\,'y'\] is the variable and \[(+b)\] is the constant.

• Equation: A statement of equality of two algebraic expression involving a variable is called an equation.
• Simple linear equation: An equation which contains only one variable of degree 1 is called a simple linear equation.                                   

e.g.,      (i) \[5x-1=6x+m\]                                  (ii) \[3\left( x-4 \right)=5\] (iii)\[2y+5=\frac{y}{6}-2\]                                  (iv) \[\frac{t-1}{6}+\frac{2t}{7}=a\]

• Solution of an equation: That value of the variable, which when substituted in the given equation, makes the two sides L.H.S. (Left Hand Side) and R.H.S. (Right hand sided) of the equation equal is called the solution or root of that equation.

e.g., \[2x+6=3x-10\Rightarrow 6+10=3x-2x\Rightarrow 16=x\] Verification Substituting \[x=16\] we have LHS \[=2\times 16+6=38\] & RHS \[=\text{ }3\times 16-10=38\] \[\therefore \]\[x=16\]is a solution of the above equation.

• Rules for solving an equation

(a) Same number can be added to both sides of an equation. (b) Same number can be subtracted from both sides of an equal. (c) Both sides of an equation can be multiplied by the same non - zero number (d) Both sides of an equation can be divided by the same non - zero number (e) Cross multiplication: If\[\frac{ax+b}{cx+d}=\frac{p}{q}\], then \[q\left( ax+b \right)=p\left( cx+d \right).\] This process is called cross multiplication.  

ALGEBRAIC EXPRESSIONS   FUNDAMENTALS

• Algebra: It is a branch of mathematics in which we use literal numbers and statements symbolically. Literal numbers can be positive or negative. They are Variables.
• Variable: A symbol which takes various values is known as a variable. Normally it is denoted by x, y, z etc.
• Coefficient: Symbols like a, b, 1, m etc.., are used to denote the coefficients. Coefficients that are numbers are called numerical coefficients.
• Algebraic expression: A combination of constants and variables connected by some or all of the four fundamental operations \[+,\text{ }-,\text{ }\times \]and \[\div \] is called an algebraic expression.

e.g., \[-5x+6\]is an algebraic expression. Here \[-\text{ }5\]is the coefficient of the variable 'x' and 6 is the constant.

• Like and unlike terms: In a given algebraic expression, the terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms.

e.g., 8xy and \[-\,4xy\]are like terms while 6xy and 6x are unlike terms,

• Factors: Each term of an algebraic expression consists of a constant or product of constant and variables.

 

• Various types of algebraic expression:

(a) Monomial: An algebraic expression which contains only one term, is called as monomial. Thus, \[2x,3y,\,\,5xy,\text{ }6a{{b}^{2}},-11\]etc. are called monomials. (b) Binomial: An algebraic expression containing two terms is called a binomial. Thus, \[\left( 2a\text{ +}\,6b \right),\left( 8-6x \right),\left( {{x}^{2}}-6x{{y}^{2}} \right),\]etc., are all binomials. (c) Trinomial: An algebraic expression containing three terms is called a trinomial. Thus, \[\left( a+2b+5c \right),\left( x+2y-3z \right),\left( {{x}^{3}}+\text{ }{{y}^{3}}+\text{ }{{z}^{3}} \right),\]etc., are all trinomials. (d) Polynomial: An expression containing two or more terms is called a polynomial.

• Addition of Algebraic Expression: While adding algebraic expressions, we collect the like terms and add them. The sum of several like terms is another like term whose coefficient is the sum of the coefficients of those like terms. The like terms are added and the unlike terms are left as they are. e.g.,

\[7x+2y\text{+}8x+3{{x}^{2}}+5{{x}^{2}}+6{{y}^{2}}\] \[=\left( 7+8 \right)x+2y+\left( 3+5 \right){{x}^{2}}+6{{y}^{2}}\]

• Subtraction of Algebraic Expression>: The difference of two like terms is a like term whose coefficients is the difference of the numerical coefficients of the two like terms.

e.g., \[7{{x}^{2}}-8{{x}^{2}}=(7-8){{x}^{2}}=-{{x}^{2}},8y-6y-2y=(8-6-2)y=0y=0\]

• Value of an algebraic expression: The value of an algebraic expression depends on the values of the variables forming the expression.
• Using algebraic expressions - Formulae and Rules: Rules and formulae in mathematics are written in a concise and general form using algebraic expression.

Thus, the area of square \[={{a}^{2}},\]where a is the length of side of a square. The general \[({{n}^{th}})\] term of a number pattern (or a sequence) is an expression in 'n'. Thus, the nth term of the number pattern 9, 19, 22, 39.........is \[(10n-1)\]  

EXPONENTS AND POWER   FUNDAMENTALS

•                       Exponential form is nothing but repeated multiplication.

There are two part of an exponent. Exponent \[\to \] base, Power/ Index Example:

•                         Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

\[a\text{ }\times \text{ }a\text{ }\times \text{ }a={{a}^{3}}\] (read as 'a' cubed or 'a' raised to the power 3) \[a\text{ }\times \text{ }a\text{ }\times \text{ }a\text{ }\times \text{ }a\text{ }\times \text{ }a\text{ }\times \text{ }a={{a}^{6}}\] (read as 'a raised to the power 6 or 6th  power of a) .............................................................................................................. \[a\text{ }\times \text{ }a\text{ }\times \text{ }a\]....... (n factors) \[=\text{ }{{a}^{n}}\](read as 'a' raise to the power n or nth  power of a)

•                        (a) When a negative number is raised to an even power the value is always positive.

e.g., \[{{(-5)}^{6}}=(-5)\times (-5)\times (-5)\times (-5)\times (-5)\times (-5)=15625\] (b) When a negative number is raised to an odd power, the value is always negative. e.g., \[{{(-3)}^{5}}=(-3)\times (-3)\times (-3)\times (-3)\times (-3)=-243\] Note: (a) \[{{(-1)}^{odd\,\,number}}=-1\]                                  (b) \[{{(-1)}^{even\,\,number}}=+1\]   Elementary question 1: In \[{{3}^{5}},\]what is the base and power respectively? Ans.:    Base = 3 Power = 5   Elementary Question 2: Write 32 in exponent form Ans.:    \[32=2\times 2\times 2\times 2\times 2={{2}^{5}}~\]     where base = 2              power / Index = 5  

•                         Laws of Exponents:

For any non-zero integers 'a' and 'b' and whole numbers 'm' and 'n', (a)\[a\times a\times a\times ~\]............. \[\times \text{ }a\](m factors) \[={{a}^{m}}~\] (b) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\] (c) \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}},\] if m > n; = 1, if m = n ; \[=\frac{1}{{{a}^{n-m}}}\]if m < n (d) \[{{({{a}^{m}})}^{n}}={{a}^{mn}}\] (e) \[{{(ab)}^{m}}={{a}^{m}}{{b}^{m}}\] (f) \[{{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}\] (g) \[a{}^\circ =1\] Most of the questions under this chapter are applications of the above formula (a) to (g). Therefore commit them to memory (not ROT memory but learn by applying).   Elementary question 3: Evaluate:          (i) \[5\times 5\times 5\]     (ii)\[{{5}^{2}}\times {{5}^{3}}\]         (iii) \[\frac{{{5}^{3}}}{{{5}^{2}}}\]                    (iv)\[{{\left( {{5}^{2}} \right)}^{3}}\]   (v)\[{{\left( 2\times 5 \right)}^{3}}\]        (vi) \[{{\left( \frac{5}{2} \right)}^{2}}\]  (vii) \[5{}^\circ \times 2{}^\circ \times 3{}^\circ \] Ans.:    (i) \[5\times 5\times 5\](three times)\[={{5}^{3}}=125\] (ii) \[{{5}^{2}}\times {{5}^{3}}=\text{ }{{5}^{2+3}}=\text{ }{{5}^{5}}=\text{ }3125\] (iii) \[\frac{{{5}^{3}}}{{{5}^{2}}}={{5}^{3-2}}={{5}^{1}}=5\] (iv) \[{{({{5}^{2}})}^{3}}={{5}^{2\times 3}}={{5}^{6}}=15625\] (v) \[{{\left( \frac{5}{2} \right)}^{2}}=\frac{{{5}^{2}}}{{{2}^{2}}}=\frac{25}{4};\]              (vi) \[{{\left( 2\times 5 \right)}^{3}}={{2}^{3}}\times {{5}^{3}}=8\times 125=1000\] (vii) \[5{}^\circ \times 2{}^\circ \times 3{}^\circ =1\times 1\times 1=1\]  

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COMPARING QUANTITIES   FUNDAMENTALS A. Ratio and Proportion

•                         Ratio is a method of comparing two quantities of the same kind by division.
•                        The symbol used to write a ratio is ':' and is read as 'is to'.
•                        A ratio is generally expressed in its simplest form.
•                        A ratio does not have any unit, it is only a numerical value.
•                        To express two terms in a ratio, they should be in the same units of measurement.
•                         When two ratios are equal, they are said to be in proportion. The symbol for proportion is ': :' and is read as 'as to'.

For e.g., 2 is to 3 as to 6 is to 9 is written as \[2:3::6:9\] or,\[\frac{2}{3}=\frac{6}{9}\]

•                         If two ratios are to be equal or to be in proportion, their product of means should be equal to the product of extremes.

Example: If \[a:b::c:d\] then the statement ad = bc, holds good.

•                         If \[a:b\] and \[b:c\] are in proportion such that \[{{b}^{2}}=ac\] then b is called the mean proportional of \[a:b\] and\[b:c\].
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LINES AND ANGLES               FUNDAMENTALS

•               Point: A point is a geometrical representation of a location, it is represented by a dot.
•               Line: A line is a set of points that extends endlessly in both the directions i.e., a line has no end points.

•               Line segment: A line segment is a part of a line. A line segment has two end points. A line segment AB is represented as AB.

•              Ray: A ray is a part of the line which has one end point (namely its starting point).

(Here 'O' is the starting point for ray OA)

•                Angle: An angle is the union of two rays with a common initial point.

The symbol of angle is\[\angle \]. An angle is measured in degrees\[({}^\circ )\]. The angle formed by the two rays \[\overline{AB}\] and \[\overline{AC}\] is denoted by \[\angle BAC\] or\[\angle CAB\].       more...

PROPERTIES OF TRIANGLE   FUNDAMENTALS

•            A triangle (denoted as A delta) is a closed figure bounded by three line segments, it has three vertices, three sides and three angles. The three sides and three angles of a triangle are called its six elements.

  Elementary Question -1 Identify triangle among following figures and also identify its six elements and vertices. The figure (iv) is a triangle Its sides are AB, BC, CA and angles are \[\angle A,\angle B,\angle C\] (also written as \[\angle BAC,\angle CBA\]and \[\angle ACB\]). These are six elements and its vertices are points A, B, C.

•          A triangle is said to be

(a) An acute angled triangle, if each one of its angles measures less than \[90{}^\circ .\] (b) A right angled triangle, if any one of its angles measures \[90{}^\circ .\] (c) An obtuse angled triangle, if any one of its angles measures more than \[90{}^\circ .\]        Note:    A triangle cannot have more than one right angle. A triangle cannot have more than one obtuse angle. In a right triangle, the sum of the acute angles is \[90{}^\circ .\]

•          Angle sum property: The sum of the angle of a triangle is \[180{}^\circ .\]

This is such an important property that it will be used right from class VII to more...

CONGRUENCE OF TRIANGLES   FUNDAMENTALS      

•          Two figures, having exactly the same shape and size are said to be congruent.
•          Two triangles are said to be congruent, if pairs of corresponding sides and corresponding angles are equal.

  Note: The symbol \[\cong \] is used to denote 'is congruent to'.

•        Two line segments are congruent, If they have the same length, i.e. \[\overrightarrow{AB}\cong \overrightarrow{CD}\] and is read as line segment \[\overrightarrow{AB}\] is congruent to the line segment\[\overrightarrow{CD}\].
•         Two angles are congruent, if they have the same measure. "\[\angle A\] is congruent to \[\angle B\]" is written symbolically as \[\angle A\cong \angle B\] or \[\angle A=\angle B\].
•         (S.S.S.) Congruence criteria: If the three sides of a triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.

e.g.,                  In the given figure, \[\Delta \,ABC=\cong \Delta \,DEF\] by S. S. S. congruence condition.

•         (S.A.S.) congruence condition: If two sides and the included angle of a triangle are respectively equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

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