Question

Provide the missing statement and reasons for the following proof:ce 2:B(12x+20)Dge 3:А(11x +23)*EGiven: ZBDA ZAProve: x=3StatementReasonS1 2BDA ZAR1. GivenS2 BDA CDER2S3 CD ZAR3 Transitive Property of CongruenceS4 m CDE-MZAR4S5R5. Substitution Property of EqualityS6 12 - 11x + 3R6S7-3R7 Subtraction Property of Equality

Answer 1

Problem Statement

We are given a figure and we are asked to prove that x = 3.

Method

To solve this problem, we simply need to take note of the properties we might need.

1. Vertically Opposite Angles:

Given the orientation of angles below:

We have the following

`\angle A=\angle B\text{ (VERTICALLY OPPOSITE ANGLES)}`

2. Transitive Property of Congruence:

This property is described below:

`\begin{gathered} \text{If X = Y, and Y = Z,} \\ \text{Then by the TRANSITIVE PROPERTY OF CONGRUENCE, we have:} \\ X=Z \end{gathered}`

3 Transitive property of Equality:

This property just shows that two magnitudes are equal. Unlike the Transitive property of congruence, that deals with angles and shapes, the Transitive Property of Equality deals with numbers (or the magnitudes of angles)

For example:

`\begin{gathered} A=B\text{ and B = 5} \\ \therefore A=5,\text{ BY THE TRANSITIVE PROPERTY OF EQUALITY} \end{gathered}`

4. Substitution Property of Equality

This property states that if two angles are equal, then either can replace the other in any equation or expression.

For example:

`\begin{gathered} C=A+B \\ \text{also,} \\ A=x,B=y \\ \therefore C=x+y,\text{ By THE SUBSTITUTION PROPERTY OF EQUALITY} \end{gathered}`

5. Subtraction Property of Equality:

This property says that if we subtract one value from one side of the equal sign, we should subtract the same value from the other side of the equal sign to keep both sides balanced and equal.

For example,

`\begin{gathered} \text{If A = A} \\ By\text{ the SUBTRACTION PROPERTY OF EQUALITY, we can say:} \\ \text{subtract 2 from both sides} \\ \\ A-2=A-2 \end{gathered}`

With these definitions, we can proceed to answer the question.

Implementation

`\begin{gathered} \angle\text{BDA}\cong\angle\text{CDE} \\ \text{Because of the VERTICALLY OPPOSITE ANGLES property} \end{gathered}`

`\begin{gathered} m\angle\text{CDE}=m\angle A \\ \text{Because of the TRANSITIVE PROPERTY OF EQUALITY} \end{gathered}`

By the SUBSTITUTION PROPERTY OF EQUALITY, we have that:

`12x+20^0=11x+23^0`

By SUBTRACTION PROPERTY OF EQUALITY, we have that:

`\begin{gathered} 12x+20^0=11x+23^0^{} \\ \text{subtract 20}^0\text{ from both sides because of the SUBTRACTION PROPERTY OF EQUALITY,} \\ \\ 12x+20^0-20^0=11x+23^0-20^0 \\ 12x=11x+3^0 \end{gathered}`

Thus, we can fill up the table as follows: