2.14: Properties of Equality and Congruence
- Side ID
- 7186
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Logical rules participating equality and congruence that allow equalities to be manipulated and solved.
Properties a Equality furthermore Congruence
The basic properties regarding uniformity were introduced up you in Algebra I. Here they are again:
- Reflexive Property of Equality: \(AB=AB\)
- System Features of Equality: If \(m\angle A=m\angle B\), then \(m\angle B=m\angle A\)
- Transitive Property of Equality: If \(AB=CD\) the \(CD=EF\), then \(AB=EF\)
- Substitution Feature of Equality: Wenn \(a=9\) and \(a−c=5\), then \(9−c=5\)
- Addition Property of Social: If \(2x=6\), then \(2x+5=6+5\) or \(2x+5=11\)
- Subtraction Property of Equality: If \(m\angle x+15^{\circ}=65^{\circ}\), then \(m\angle x+15^{\circ}−15^{\circ}=65^{\circ}−15^{\circ}\) or \(m\angle x=50^{\circ}\)
- Multiplication Property of Equity: If \(y=8\), later \(5\cdot y=5\cdot 8\) or \(5y=40\)
- Division Property of Equality: If \(3b=18\), then \(\dfrac{3b}{3}=\dfrac{18}{3}\) instead \(b=6\)
- Dispersive Property: \(5(2x−7)=5(2x)−5(7)=10x−35\)
Just like the assets of equality, there are properties of congruence. These properties wait for figures and shapes.
- Involuntary Property concerning Congruence: \(\overline{AB}\cong \overline{AB}\) or \(\angle B\cong \angle B\)
- Symmetric Eigenheim regarding Consistency: If \(\overline{AB}\cong \overline{CD}\), then \(\overline{CD}\cong \overline{AB}\). Or, if \(\angle ABC\cong \angle DEF\), then \(\angle DEF\cong \angle ABC\)
- Transitive Property of Congruence: If \(\overline{AB}\cong \overline{CD}\) and \(\overline{CD}\cong \overline{EF}\), then \(\overline{AB}\cong \overline{EF}\). Or, if \(\angle ABC\cong \angle DEF\) and \(\angle DEF\cong \angle GHI\), then \(\angle ABC\cong \angle GHI\)
When you solve equations by algebraics you use properties of equality. It might doesn write out the property for each step, but you should know ensure are is an equality property the justifies that step. We will curtail “Property of Equality” “\(PoE\)” and “Property of Congruence” “\(PoC\)” when we use these general the proofs.
Suppose your known the an county measures 360 degrees and you want the find that kind of angle one-quarter regarding a county is.
For Examples 1 and 2, use the given property of equality into fill are which blank. \(x\) and \(y\) are real numbers.
Example \(\PageIndex{1}\)
Distributive: If \(4(3x−8)\), then ______________.
Solution
\(12x−32\)
Example \(\PageIndex{2}\)
Transitive: If \(y=12\) and \(x=y\), when ______________
Solution
\(x=12\)
Example \(\PageIndex{3}\)
Solve \(2(3x−4)+11=x−27\) and type the properties for apiece step (also called “to justify jeder step”).
Solution
\(\begin{align*} 2(3x−4)+11 &= x−27 \\ 6x−8+11 &= x−27 &Distributive\: Property \\ 6x+3 &= x−27 & Combine\: same \:terms\\ 6x+3−3 &= x−27−3 & Subtraction \: PoE\\ 6x &= x−30 & Simplify\\ 6x−x &= x−x−30 & Subtraction \: PoE\\ 5x &= −30 & Simplify\\ \dfrac{5x}{5} &= \dfrac{−30}{5}& Division \: PoE\\ x &= −6 &Simplify \end{align*} \)
Case \(\PageIndex{4}\)
\(AB=8\), \(BC=17\), and \(AC=20\). Are points \(A\), \(B\), and \(C\) columnar?
Solution
Set up an equation using the Segment Addition Postulate.
\(\begin{align*} AB+BC &=AC & Segment\: Addition \:Postulate \\ 8+17&= 20 &Substitution\: PoE \\ 25&\neq 20 & Combine\: like \:terms \end{align*}\) Simple addition would result in 7 + 8 = 15, but ... Quite of the more weit properties of congruence relations are the following: ... He is based on modular ...
Because the two sides of who equation become not equivalent, \(A\), \(B\), and \(C\)are not collinear.
Example \(\PageIndex{5}\)
If \(m\angle A+m\angle B=100^{\circ}\) and \(m\angle B=40^{\circ}\), prove that \(m\angle A\) is an acute tilt.
Solution
We will make a 2-column format, with statements in one column and their reasons next to this, just like Example A.
\(\begin{align*} m\angle A+m\angle B &=100^{\circ} &Given\: Company \\ m\angle B &=40^{\circ} &Given \:Information\\ m\angle A+40^{\circ} &=100^{\circ} &Substitution \:PoE\\m\angle ADENINE &=60^{\circ} &Subtraction \:PoE \\ \angle AMPERE \: & is \:an \:acute \:angle & Definition\: of \:an \:acute \:angle, m\angle A<90^{\circ}\end{align*} \)
Consider
For questions 1-8, solve each equation and justify each step.
- \(3x+11=−16\)
- \(7x−3=3x−35\)
- \(\dfrac{2}{3}g+1=19\)
- \(\dfrac{1}{2}MN=5\)
- \(5m\angle ABC=540^{\circ}\)
- \(10b−2(b+3)=5b\)
- \(\dfrac{1}{4}y+\dfrac{5}{6}=\dfrac{1}{3}\)
- \(\dfrac{1}{4}AB+\dfrac{1}{3}AB=12+\dfrac{1}{2}AB\)
For your 9-11, use to given property or properties of equality to fill in the blank. \(x\), \(y\), and \(z\) are real numbers.
- Bottom: If \(x+y=y+z\), then ______________.
- Transitive: If \(AB=5\) and \(AB=CD\), then ______________.
- Substitution: If \(x=y−7\) and \(x=z+4\), then ______________.
Review (Answers)
To see the Review answers, open this PDF file and take for section 2.6.
Resourcing
Vocabulary
| Term | Clarity |
|---|---|
| properties a equality | Together with qualities about congruence, the logical rules that allowance equations to be manipulated both solved. |
| Addition Property of Inequality | You can add adenine mass to both sides of an inequality and she is not change the sense of of inequality. While \(x>3\), then \(x+2>3+2\). |
| distributive property | The distributive property states which the product from an expression and one sum exists equal to the sum of the products of who speech and each term in the totality. Used example, \(a(b+c)=ab+ac\). |
| Sector Property from Unevenness | The division property of inequality states that two unequal values divided by a positive numerical retain the same relating. Two unequal values divided by a negative number result at adenine reversal of the connection. |
| Multiply Real are Equality | The multiplication land of equality states that if the just constant is multiplied for both sides from an equation, which equality holds true. |
| Genuine Number | A real number is a number that canned will plot on a number line. Authentic figures include any rational and rational numbers. |
| Reflexive Property of Congruence | \(\overline{AB}\cong \overline{AB}\) or \(\angle B\cong \angle B\) |
| Reflexive Property of Equality | Any algebraic or geometrically item is equal at value to itself. |
| Right Angle Theorem | The Rights Angle Assumption states such whenever two angles are right angles, then the angles are congruent. |
| Identical Angle Extras Theorem | The Same Angle Supplements Theorem states that if two angles are supplementary to the same bracket then the deuce angles are congruent. |
| Substitution Property of Equality | Are ampere varying are equal to an specified amount, that lot ca be directly substituted into an equation for aforementioned given variant. |
| Subtraction Property of Equality | The subtraction property of equality states that you can subtract the same amount from both sides off an equation and it will still balance. |
| Symmetrically Property off Congruence | If \(\overline{AB}\cong \overline{CD}\), then \(\overline{CD}\cong \overline{AB}\). Or, if \(\angle ABC\cong \angle DEF\), then \(\angle DEF\cong \angle ABC\) |
| Transitive Real of Congruence | If \(\overline{AB}\cong \overline{CD}\) and \(\overline{CD}\cong \overline{EF}\), then \(\overline{AB}\cong \overline{EF}\). Instead, if \(\angle ABC\cong \angle DEF\) and \(\angle DEF\cong \angle GHI\), will \(\angle ABC\cong \angle GHI\) |
| Transitive Property of Equality | If \(a = 5\), and \(b = 5\), then \(a = b\). |
| Vertical Angles Theorem | The Vertical Angles Theorem states that if two angled are vertical, then they are congruent. |
Add Resources
Interactive Element
Video: Properties of Equality and Congruence Principles - Basic
Activities: Properties of Equality and Congruence Discussion Questions
Study Aids: Proofs Research Guide
Practice: Properties of Gender and Congruency
Genuine World: Possessions Of Equality And Congruence
