11. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #2 using the picture.
12. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #3 using the picture.
13. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #4 using the picture.
14. ( overline{ab} cong overline{bc} ) and ( overline{bc} cong overline{fg} ), then ____ fill in the blank, hen________\
15. image: angle 1 and 2 form a linear pair angle 1 and angle 2 form a linear pair. if the measure of angle 1 = ( (2x - 4) ) and the measure of angle 2 = ( (2x + 4) ), what is the equation for solving ( x )?
16. image: ray ( bd ) bisects ( angle abc ), ( angle abd = 3x + 7 ), ( angle dbc = 4x - 17 ) given ray ( bd ) bisects ( angle abc ), what is the equation for solving ( x )?
17. image: ( m ) is between ( c ) and ( g ), ( cm = x + 4 ), ( mg = 2x ), ( cg = 4x + 2 ) ( m ) is between ( c ) and ( g ). if ( cm = x + 4 ), ( mg = 2x ), and ( cg = 4x + 2 ), what is the equation for solving ( x )?
18. image: three block figures what is the pattern?

Question

11. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #2 using the picture.
12. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #3 using the picture.
13. 1. ( 3(x + 1) = 2x + 7 ) 1. given 2. ( 3x + 3 = 2x + 7 ) 2. ______ 3. ( x + 3 = 7 ) 3. ______ 4. ( x = 10 ) 4. ______ find the algebraic property for step #4 using the picture.
14. ( overline{ab} cong overline{bc} ) and ( overline{bc} cong overline{fg} ), then ____ fill in the blank, 	hen________\
15. image: angle 1 and 2 form a linear pair angle 1 and angle 2 form a linear pair. if the measure of angle 1 = ( (2x - 4) ) and the measure of angle 2 = ( (2x + 4) ), what is the equation for solving ( x )?
16. image: ray ( bd ) bisects ( angle abc ), ( angle abd = 3x + 7 ), ( angle dbc = 4x - 17 ) given ray ( bd ) bisects ( angle abc ), what is the equation for solving ( x )?
17. image: ( m ) is between ( c ) and ( g ), ( cm = x + 4 ), ( mg = 2x ), ( cg = 4x + 2 ) ( m ) is between ( c ) and ( g ). if ( cm = x + 4 ), ( mg = 2x ), and ( cg = 4x + 2 ), what is the equation for solving ( x )?
18. image: three block figures what is the pattern?

Accepted Answer

### Question 14
# Brief Explanations:
This is about the transitive property of congruence. If $\overline{AB} \cong \overline{BC}$ and $\overline{BC} \cong \overline{FG}$, then by the transitive property of congruence, $\overline{AB} \cong \overline{FG}$.
# Answer:
$\overline{AB} \cong \overline{FG}$

### Question 15
# Explanation:
## Step1: Recall linear pair property
A linear pair of angles sums to $180^\circ$. So, $m\angle1 + m\angle2 = 180^\circ$.
## Step2: Substitute angle measures
Given $m\angle1=(2x - 4)$ and $m\angle2=(2x + 4)$, substitute into the equation: $(2x - 4)+(2x + 4)=180$.
# Answer:
$(2x - 4)+(2x + 4)=180$

### Question 16
# Explanation:
## Step1: Recall angle bisector definition
If ray $BD$ bisects $\angle ABC$, then $\angle ABD=\angle DBC$.
## Step2: Substitute angle expressions
Given $\angle ABD = 3x + 7$ and $\angle DBC=4x - 17$, the equation is $3x + 7=4x - 17$.
# Answer:
$3x + 7 = 4x - 17$

### Question 17
# Explanation:
## Step1: Recall segment addition postulate
If $M$ is between $C$ and $G$, then $CM + MG=CG$.
## Step2: Substitute segment lengths
Given $CM=x + 4$, $MG = 2x$, and $CG=4x + 2$, substitute into the postulate: $(x + 4)+2x=4x + 2$.
# Answer:
$(x + 4)+2x=4x + 2$

### Question 18
# Brief Explanations:
Looking at the figures:
- First figure: Rows - 2 rows (top 1, bottom 2), columns - 2 columns (left 2, right 1). Total squares: $1 + 2=3$ or $2+1 = 3$.
- Second figure: Rows - 3 rows (top 1, middle 2, bottom 3), columns - 3 columns (left 3, middle 2, right 1). Total squares: $1+2 + 3=6$ or $3 + 2+1=6$.
- Third figure: Rows - 4 rows (top 1, then 2, then 3, then 4), columns - 4 columns (left 4, then 3, then 2, then 1). Total squares: $1+2+3 + 4 = 10$ or $4+3+2 + 1=10$.

The pattern is that for the $n$-th figure (where $n = 1,2,3$ for the given figures), the number of squares is the sum of the first $n + 1$ positive integers (or the sum of integers from $1$ to $n + 1$) or equivalently the sum of integers from $1$ to $k$ where $k=n + 1$, and also it can be seen as the sum of a triangular number pattern where each figure has a number of squares equal to $\frac{(n + 1)(n + 2)}{2}$ (since sum of first $k$ integers is $\frac{k(k + 1)}{2}$, here $k=n + 1$). Visually, each subsequent figure has one more row and one more column of squares added in a triangular (staircase) manner, with the number of squares increasing by $n + 2$ (for $n$-th figure) from the previous. For example, from first to second: $3$ to $6$ (increase by $3$), second to third: $6$ to $10$ (increase by $4$), which follows the pattern of the $n$-th triangular number where the difference between consecutive triangular numbers is $n + 1$.
# Answer:
The pattern is that the $n$-th figure (for $n = 1,2,3,\dots$) has a number of squares equal to the sum of the first $n + 1$ positive integers (or $\frac{(n + 1)(n + 2)}{2}$), and the figures form a staircase - like pattern where each figure has one more row and one more column of squares than the previous, with the number of squares increasing as triangular numbers ($3,6,10,\dots$ for $n = 1,2,3$ respectively).

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QUESTION IMAGE

Question

Explanation:

Question 14

Step1: Recall linear pair property

Step2: Substitute angle measures

Step1: Recall angle bisector definition

Step2: Substitute angle expressions

Answer:

Question 15