Grade 6 Workbook — Chapter 15: Congruence and Regular Polygons

SOL 6.MG.4 · Companion to Textbook Chapter 15

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 15 · Congruence and Regular Polygons

Standard 6.MG.4

In this chapter you will:

• Decide when segments and angles are congruent
• Identify regular polygons
• Draw lines of symmetry that split a regular polygon into two congruent parts
• Decide whether two polygons are congruent or noncongruent

Words to know: congruent · noncongruent · similar · polygon · vertex · regular polygon · line of symmetry · corresponding sides · corresponding angles

Notation bar: $\overline{AB}$ is a segment · $AB$ is its length · $\angle ABC$ is an angle · $m\angle ABC$ is its measure · $\cong$ joins figures · $=$ joins numbers · $\ncong$ means not congruent

PAGE 2 — Congruent segments and angles

15.1 Congruent Segments and Angles

FIGURE: fig1-congruent-segments.png (full width) [Three segments in different orientations: AB horizontal 3 cm, CD slanted 3 cm, EF vertical 5 cm, with one tick mark on AB and CD.]

Complete the sentences.

Two segments are congruent when they have the same ____________ .

Two angles are congruent when they have the same ____________ .

Congruent means same ____________ and same ____________ .

Congruent or not congruent? Write $\cong$ or $\ncong$ between each pair.

Fill in from the given congruence.

$\overline{PQ} \cong \overline{RS}$ and $RS = 21$ mm, so $PQ =$ ______

$\angle D \cong \angle E$ and $m\angle E = 47°$, so $m\angle D =$ ______

$\angle R \cong \angle S$ and $m\angle S = 64°$, so $m\angle R =$ ______

PAGE 3 — Notation and marks

Reading the Marks

FIGURE: fig2-congruent-angles.png (half width) [Two 45-degree angles, one with short rays and one with long rays, each marked with a single arc.]

Both angles measure $45°$. Are they congruent? ______

Why don't the longer rays make a bigger angle?

FIGURE: fig3-tick-marks.png (half width) [Quadrilateral PQRS with one tick on PQ and RS, two ticks on QR and SP, one arc at P and R, two arcs at Q and S.]

List every congruent pair you can read from the marks.

Segments: ____________ and ____________ ; ____________ and ____________

Angles: ____________ and ____________ ; ____________ and ____________

$\cong$ or $=$? Circle the correct symbol in each statement.

Cut it in half. A ribbon $25$ cm long is cut into two congruent pieces. Each piece: ______ cm

A board $18$ cm long is cut into two congruent pieces. Each piece: ______ cm

PAGE 4 — Exit ticket 15.1

Exit Ticket · Lesson 15.1

Name: ________________________ Date: ____________

1. $AB = 13$ cm and $CD = 13$ cm. Write a congruence statement. _______________

2. $\angle R \cong \angle S$ and $m\angle S = 64°$. Find $m\angle R$. ______

3. Are angles of $25°$ and $52°$ congruent? ______ Why?

4. Explain the difference between $\cong$ and $=$, with one example of each.

PAGE 5 — Regular polygons

15.2 Regular Polygons

FIGURE: fig4-regular-polygons.png (full width) [An equilateral triangle, square, regular pentagon, and regular hexagon in a row, each with matching tick marks on every side and matching arcs at every angle, labeled with 60, 90, 108, and 120 degrees.]

A regular polygon has all sides ____________ AND all angles ____________ .

Both conditions must be true. One failure is enough to make a polygon not regular.

Complete the name table.

Complete the angle table.

PAGE 6 — Regular or not?

Two Ways to Fail

FIGURE: fig5-not-regular.png (full width) [A 3-by-7 rectangle with right-angle marks and side labels, next to a rhombus with all sides 6 and angles 70, 110, 70, 110 degrees; both labeled not regular.]

Which test does each figure fail? Write "sides" or "angles."

Rectangle $3$ by $7$: fails the ____________ test.

Rhombus with $70°$ and $110°$ angles: fails the ____________ test.

Regular or not regular? Give a reason.

Perimeter of a regular polygon: $P = n \times s$

Counterexample. Kai says all congruent sides is enough to be regular. Draw or describe a figure that proves him wrong.

PAGE 7 — Exit ticket 15.2

Exit Ticket · Lesson 15.2

Name: ________________________ Date: ____________

1. What two conditions make a polygon regular?

2. Name the polygon with $9$ sides. ______________

3. Regular hexagon with $9$ cm sides. Perimeter: ______

4. Why are most rectangles not regular? Which kind is?

PAGE 8 — Lines of symmetry

15.3 Lines of Symmetry

A line of symmetry folds a figure into two parts that match exactly. The two parts are ____________ .

FIGURE: fig6-symmetry-odd.png (half width) [Equilateral triangle with 3 dashed lines of symmetry from each vertex to the opposite side midpoint, and a regular pentagon with its 5 dashed lines drawn the same way.]

FIGURE: fig7-symmetry-even.png (half width) [Square with 4 dashed lines of symmetry (2 diagonals, 2 through side midpoints) and a regular hexagon with 6 dashed lines (3 vertex to vertex, 3 midpoint to midpoint).]

Draw all lines of symmetry on each figure, then count.

The pattern: a regular polygon has ____________ lines of symmetry as it has sides.

PAGE 9 — Odd, even, and the exceptions

Where the Lines Go

Sort the two cases.

Odd number of sides: every line runs from a ____________ to the ____________ of the opposite side.

Even number of sides: half the lines join opposite ____________ , and half join the ____________ of opposite sides.

Complete the table.

Figures with few or no lines of symmetry.

Rectangle $3$ cm by $7$ cm: ______ lines. Why aren't the diagonals lines of symmetry?

Parallelogram that is not a rectangle or rhombus: ______ lines

Scalene triangle: ______ lines

Work backward. A regular polygon has $15$ lines of symmetry. Sides: ______

Are its lines vertex to vertex, vertex to side midpoint, or a mix? ____________ Why?

Apply it. A designer folds a paper regular hexagon along a line of symmetry and cuts two matching quilt pieces.

Why must the pieces be congruent? _______________________________________________

How many different folds would work? ______

PAGE 10 — Exit ticket 15.3

Exit Ticket · Lesson 15.3

Name: ________________________ Date: ____________

1. Lines of symmetry in a square: ______

2. Lines of symmetry in a regular decagon: ______

3. What is always true about the two parts a line of symmetry creates?

4. Why does a regular polygon have exactly as many lines of symmetry as sides?

PAGE 11 — Corresponding parts

15.4 Congruent and Noncongruent Polygons

FIGURE: fig8-congruent-triangles.png (full width) [Triangle ABC with AB = 6 cm, BC = 8 cm, AC = 10 cm and angle marks, beside triangle DEF rotated and flipped with matching side lengths, tick marks, and arcs.]

Two polygons are congruent when every pair of corresponding ____________ is congruent AND every pair of corresponding ____________ is congruent.

From $\triangle ABC \cong \triangle DEF$, fill in all six facts.

Order matters. From $ABCD \cong WXYZ$: the side matching $\overline{BC}$ is ______ and the angle matching $\angle D$ is ______

From $PQRS \cong TUVW$: the side matching $\overline{RS}$ is ______ and the angle matching $\angle Q$ is ______

Fill in the measurements. $\triangle ABC \cong \triangle DEF$, $AB = 9$ m, $BC = 5$ m, $AC = 11$ m, $m\angle B = 100°$.

$DE =$ ______ $EF =$ ______ $DF =$ ______ $m\angle E =$ ______

PAGE 12 — Same shape is not enough

Congruent vs. Same Shape

FIGURE: fig9-same-shape-not-congruent.png (half width) [Two squares side by side, one with all sides 4 cm and one with all sides 9 cm, both with right-angle corner marks.]

Both figures are squares. Complete the analysis.

All corresponding angles congruent? ______

All corresponding sides congruent? ______

Congruent or noncongruent? ____________

Same shape, different size, is called ____________ .

Congruent means same shape AND same size.

Congruent or noncongruent? Give the reason.

PAGE 13 — Quality control

Congruent or Not?

1. Turning and flipping. A triangle is turned upside down and set beside an identical triangle. Congruent? ______

What does turning a figure change? ____________ What does it not change? ____________

2. Regular hexagons. Two regular hexagons each have $3$ cm sides. Congruent? ______

How does the word regular make this quick to decide?

3. Factory plates. A template pentagon has sides $4$ cm, $4$ cm, $6$ cm, $6$ cm, $5$ cm with matching angles. A plate comes out $4$ cm, $4$ cm, $6$ cm, $6$ cm, $5.2$ cm.

Congruent? ______ Which side is wrong? ______ By how much? ______

What must change? _______________________________________________

4. Shelf brackets. Template triangle: $8$ in, $15$ in, $17$ in. Bracket A: $8$ in, $15$ in, $17$ in. Bracket B: $8$ in, $15$ in, $16$ in.

Congruent bracket: ______ What is wrong with the other? _______________________________________________

5. Explain. A classmate says two polygons must be congruent if all corresponding angles are congruent. Give a counterexample with measurements and name the correct word for the classmate's figures.

PAGE 14 — Exit ticket 15.4

Exit Ticket · Lesson 15.4

Name: ________________________ Date: ____________

1. Two rectangles each measure $5$ in by $8$ in, corresponding parts matching. Congruent or noncongruent? ____________

2. $\triangle ABC \cong \triangle XYZ$ and $m\angle C = 35°$. Find $m\angle Z$. ______

3. Squares with $4$ cm and $9$ cm sides. Same shape? ______ Congruent? ______

4. What must be true about the sides and angles of two congruent polygons?

PAGE 15 — Chapter 15 review, part 1

Chapter 15 Review

Part A · Regular polygons

1. Two conditions for a regular polygon: _______________________________________________

2. Regular or not, with a reason: a) square, $7$ cm sides ____________ b) rectangle $5$ m by $11$ m ____________ c) rhombus $9$ in sides, $60°$ and $120°$ ____________ d) equilateral triangle, $2$ ft sides ____________

3. Name the regular polygon with a) $6$ sides ____________ b) $8$ sides ____________ c) $10$ sides ____________

4. Regular pentagon, $11$ cm sides. Perimeter: ______

5. Regular octagon, perimeter $56$ in. Side: ______

Part B · Lines of symmetry

6. Lines of symmetry: a) equilateral triangle ______ b) square ______ c) regular hexagon ______ d) regular decagon ______

7. Where does each line of symmetry of a regular pentagon go, and why does none join two vertices?

   ---

8. The regular octagon's $8$ lines split into two kinds. Describe each and give how many: _______________

9. What is always true about the two parts a line of symmetry creates? _______________

10. Rectangle $4$ cm by $10$ cm: ______ lines. Why not the diagonals? _______________

PAGE 16 — Chapter 15 review, part 2

Chapter 15 Review (continued)

Part C · Congruent segments and angles

11. $AB = 15$ mm and $CD = 15$ mm. Congruence statement: _______________

12. $\angle J \cong \angle K$, $m\angle K = 118°$. $m\angle J =$ ______

13. Are $61°$ and $16°$ congruent? ______ Why? _______________

14. When do you use $\cong$ and when $=$? Give one example of each: _______________

Part D · Congruent and noncongruent polygons

15. Congruent or noncongruent, with a reason: a) two rectangles, each $6$ cm by $9$ cm ______ b) rectangles $6$ cm by $9$ cm and $6$ cm by $8$ cm ______ c) two regular hexagons, $5$ mm sides ______ d) squares with $3$ in and $12$ in sides ______

16. $\triangle ABC \cong \triangle RST$, $AB = 12$ cm, $AC = 7$ cm, $m\angle B = 45°$. $RS =$ ______ $RT =$ ______ $m\angle S =$ ______

17. $PQRS \cong TUVW$. Side matching $\overline{RS}$: ______ Angle matching $\angle Q$: ______

Part E · Application and reasoning

18. Regular hexagon tile, $4$ cm sides. Perimeter: ______ Lines of symmetry: ______

    Why is any two of these tiles congruent? _______________________________________________

19. Template triangle $8$ in, $15$ in, $17$ in. Bracket A: $8$, $15$, $17$. Bracket B: $8$, $15$, $16$.

    Congruent bracket: ______ Problem with the other: _______________________________________________

20. Counterexample to "all corresponding angles congruent means congruent":

    ---

    Correct word for those figures: ____________

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Notation bar: repeat the page 1 notation bar as a narrow footer strip on pages 2–4 and 11–14
• Draw-on figures: page 8 needs clean unmarked outlines of the seven regular polygons, large enough (about 1.5 in across) for students to draw symmetry lines by hand; supply these as outline-only versions of fig4-regular-polygons.png
• Figure widths: fig1-congruent-segments.png, fig4-regular-polygons.png, fig5-not-regular.png, and fig8-congruent-triangles.png full width; the rest half width
• Tick marks: print tick marks and arcs heavy enough to survive black-and-white photocopying
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow