20. to prove that the side-angle-side congruence criterion is true using transformations, start with the given information $\angle a \cong \angle d$ and $\overline{ab} \cong \overline{de}, \overline{ac} \cong \overline{df}$ $\overline{ab} \cong \overline{de}, \overline{bc} \cong \overline{ef}$ $\overline{ab} \cong \overline{de}, \angle b \cong \angle e$ and prove $\triangle abc \cong \triangle def$. next, apply a translation to $\triangle abc$ so that the image of $a$ is $d$ $e$ $f$ 21. select all that apply about the reflection in the figure below. figure of triangle reflection a. $a = a$ b. $\overline{ac} \overline{ac}$ c. line $m$ is the perpendicular bisector of $\overline{bb}$ d. $\angle abc \cong \angle abc$ e. $\overline{bb} \cong \overline{cc}$ 22. if $m\angle nop = 24$ and $m\angle noq = 110$, what is $m\angle poq$? figure of angle with points n, o, p, q 23. which theorem can you use to prove that $\triangle ghj$ and $\triangle gkj$ are congruent? figure of two right triangles sharing side gj a asa c sss b sas d hl 24. points a, b, and c are collinear. if $bc = 8.5$ and $ac = 13.2$, select all possible values of $ab$. a. 4.7 b. 3.8 c. 21.7 d. 8.5 e. 13.2 25. match each figure with the types of symmetries it has. figures: parallelogram, pentagon, cross reflectional rotational a b c

Question

20. to prove that the side-angle-side congruence criterion is true using transformations, start with the given information $\angle a \cong \angle d$ and $\overline{ab} \cong \overline{de}, \overline{ac} \cong \overline{df}$ $\overline{ab} \cong \overline{de}, \overline{bc} \cong \overline{ef}$ $\overline{ab} \cong \overline{de}, \angle b \cong \angle e$ and prove $\triangle abc \cong \triangle def$. next, apply a translation to $\triangle abc$ so that the image of $a$ is $d$ $e$ $f$ 21. select all that apply about the reflection in the figure below. figure of triangle reflection a. $a = a$ b. $\overline{ac} > \overline{ac}$ c. line $m$ is the perpendicular bisector of $\overline{bb}$ d. $\angle abc \cong \angle abc$ e. $\overline{bb} \cong \overline{cc}$ 22. if $m\angle nop = 24$ and $m\angle noq = 110$, what is $m\angle poq$? figure of angle with points n, o, p, q 23. which theorem can you use to prove that $\triangle ghj$ and $\triangle gkj$ are congruent? figure of two right triangles sharing side gj a asa c sss b sas d hl 24. points a, b, and c are collinear. if $bc = 8.5$ and $ac = 13.2$, select all possible values of $ab$. a. 4.7 b. 3.8 c. 21.7 d. 8.5 e. 13.2 25. match each figure with the types of symmetries it has. figures: parallelogram, pentagon, cross reflectional rotational a b c

Accepted Answer

### Question 22:
# Explanation:
## Step1: Identify Angle Relationship
We know that $ m\angle NOQ = m\angle NOP + m\angle POQ $. So, to find $ m\angle POQ $, we can rearrange the formula as $ m\angle POQ = m\angle NOQ - m\angle NOP $.

## Step2: Substitute Values
Given $ m\angle NOP = 24^\circ $ and $ m\angle NOQ = 110^\circ $, substitute these values into the formula:
$ m\angle POQ = 110^\circ - 24^\circ = 86^\circ $.

# Answer:
$ 86^\circ $

### Question 23:
# Brief Explanations:
To prove $ \triangle GHJ \cong \triangle GKJ $, we analyze the triangles. Both share side $ GJ $ (common side). $ \angle GBH $ and $ \angle GBK $ are right angles (given by the right - angle symbols), so $ \angle GHJ $ and $ \angle GKJ $ are right angles, meaning $ \angle GHJ\cong\angle GKJ $. Also, $ GH = GK $ (marked with the same number of ticks, indicating congruent sides). So we have a right - triangle congruence case: Hypotenuse - Leg (HL) theorem, where the hypotenuse $ GJ $ is common and one leg ($ GH\cong GK $) is congruent.

# Answer:
D. HL

### Question 24:
# Explanation:
## Step1: Analyze Collinear Points
Points $ A $, $ B $, and $ C $ are collinear. There are two cases:
- Case 1: $ B $ is between $ A $ and $ C $. Then $ AC=AB + BC $, so $ AB = AC - BC $.
- Case 2: $ C $ is between $ A $ and $ B $. Then $ AB=AC + BC $.

## Step2: Calculate Possible Values
- For Case 1: $ AB = 13.2−8.5 = 4.7 $.
- For Case 2: $ AB = 13.2 + 8.5=21.7 $.

So the possible values of $ AB $ are $ 4.7 $ and $ 21.7 $.

# Answer:
A. 4.7, C. 21.7

### Question 21:
# Brief Explanations:
- Option A: Since $ A = A' $, this is true as the reflection of $ A $ over line $ m $ is itself (from the diagram, $ A $ lies on line $ m $, so its reflection is itself).
- Option B: In a reflection, the length of a segment and its image are congruent, so $ \overline{AC}\cong\overline{A'C'} $, so $ \overline{AC}>\overline{A'C'} $ is false.
- Option C: By the property of reflection, the line of reflection (line $ m $) is the perpendicular bisector of the segment joining a point and its image (here, $ BB' $), so this is true.
- Option D: Reflections preserve angle measures, so $ \angle ABC\cong\angle A'B'C' $, this is true.
- Option E: There is no reason for $ \overline{BB'}\cong\overline{CC'} $ as the distances from $ B $ and $ C $ to the line of reflection may be different (from the diagram, the markings for $ BB' $ and $ CC' $ are not the same), so this is false.

# Answer:
A, C, D

### Question 25:
# Brief Explanations:
- **Figure A (Parallelogram)**:
    - Reflectional Symmetry: A parallelogram has no reflectional symmetry (it does not have a line that can divide it into two congruent mirror - image halves).
    - Rotational Symmetry: A parallelogram has rotational symmetry of order 2 (it can be rotated by $ 180^\circ $ about its center and map onto itself).
- **Figure B (Arrow - shaped figure)**:
    - Reflectional Symmetry: It has a vertical line of reflection (if we draw a vertical line through the center, the left and right halves are mirror images).
    - Rotational Symmetry: It does not have rotational symmetry (rotating it by less than $ 360^\circ $ will not map it onto itself).
- **Figure C (Cross - shaped figure)**:
    - Reflectional Symmetry: It has multiple lines of reflection (horizontal, vertical, and two diagonal lines) that divide it into congruent mirror - image halves.
    - Rotational Symmetry: It has rotational symmetry of order 4 (it can be rotated by $ 90^\circ $, $ 180^\circ $, $ 270^\circ $ about its center and map onto itself).

So for the table:
- Reflectional:
    - A: No (uncheck)
    - B: Yes (check)
    - C: Yes (check)
- Rotational:
    - A: Yes (check)
    - B: No (uncheck)
    - C: Yes (check)

# Answer:
|  | A | B | C |
| --- | --- | --- | --- |
| Reflectional |  | $\checkmark$ | $\checkmark$ |
| Rotational | $\checkmark$ |  | $\checkmark$ |

### Question 20:
# Explanation:
## Step1: Recall SAS Criterion
The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. We are given $ \angle A\cong\angle D $. For SAS, we need two sides that include $ \angle A $ and $ \angle D $ respectively. So we need $ \overline{AB}\cong\overline{DE} $ and $ \overline{AC}\cong\overline{DF} $ (since $ \angle A $ is included between $ \overline{AB} $ and $ \overline{AC} $, and $ \angle D $ is included between $ \overline{DE} $ and $ \overline{DF} $).

## Step2: Translation of Point $ A $
When we apply a translation to $ \triangle ABC $ to prove SAS using transformations, we want to map $ \angle A $ to $ \angle D $. So we translate $ A $ to $ D $, so that the image of $ A $ is $ D $.

# Answer:
- For the first box: $ \overline{AB}\cong\overline{DE},\overline{AC}\cong\overline{DF} $ (the first option in the first set of boxes)
- For the second box: $ D $ (the first option in the second set of boxes)

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QUESTION IMAGE

Question

Explanation:

Question 22:

Step1: Identify Angle Relationship

Step2: Substitute Values

Step1: Analyze Collinear Points

Step2: Calculate Possible Values

Answer:

Question 23: