Appendix A — Answer Key, Chapter 15: Congruence and Regular Polygons

SOL 6.MG.4 · Covers textbook Chapter 15 and the companion workbook. Item numbers match the textbook; where the workbook repeats a textbook problem, this key serves both, and workbook-only items appear in the final section. Reasoning answers show an acceptable response, not the only wording.

Lesson 15.1 — Congruent Segments and Angles

Guided practice

1. Yes. $\overline{AB} \cong \overline{CD}$, because $AB = CD = 9$ cm.
2. $\angle X \cong \angle Y$
3. No. $88 \ne 98$, so the measures are different and $\angle G \ncong \angle H$.
4. $LM = 6.5$ in
5. Use $\cong$ between figures, such as $\overline{AB} \cong \overline{CD}$ or $\angle P \cong \angle Q$. Use $=$ between numbers, such as $AB = CD$ or $m\angle P = m\angle Q$.

Independent practice

6. a) congruent b) not congruent c) congruent d) not congruent
7. $PQ = 21$ mm
8. $m\angle D = 47°$
9. Segments: $\overline{PQ} \cong \overline{RS}$ (one tick each) and $\overline{QR} \cong \overline{SP}$ (two ticks each). Angles: $\angle P \cong \angle R$ (one arc each) and $\angle Q \cong \angle S$ (two arcs each).
10. $25 \div 2 = 12.5$ cm each
11. No, the boards are not congruent, because $32 \ne 31.5$. To make them congruent the second board must be replaced or the first must be trimmed to match; either way $0.5$ in of length is the difference. Trimming the $32$ in board down to $31.5$ in removes $0.5$ in and makes the pair congruent.
12. Yes, they are congruent, because $m\angle M = m\angle N = 40°$. An angle measures the amount of turn between two rays, not how long the rays are drawn. Extending the rays makes the picture larger without opening the angle any wider.

Exit ticket 15.1

1. $\overline{AB} \cong \overline{CD}$
2. $m\angle R = 64°$
3. No. Congruent angles must have equal measures, and $25 \ne 52$.
4. $\cong$ is used between figures and $=$ is used between numbers. Example of $\cong$: $\angle P \cong \angle Q$. Example of $=$: $m\angle P = m\angle Q$. Writing $\overline{AB} = 5$ cm mixes them, because a segment is a figure and $5$ cm is a number; write $AB = 5$ cm.

Lesson 15.2 — Regular Polygons

Guided practice

1. All sides must be congruent, and all angles must be congruent. Both at the same time.
2. Yes. A square has four congruent sides and four congruent $90°$ angles, so it passes both tests.
3. Not regular. All four angles are $90°$, but the sides are $4$, $9$, $4$, $9$, and $4 \ne 9$, so the side test fails.
4. $P = 8 \times 5 = 40$ cm
5. $s = 70 \div 10 = 7$ m

Independent practice

6. a) Regular. All three sides congruent and all three angles $60°$. b) Not regular. The sides are congruent but $70 \ne 110$, so the angle test fails. c) Regular. Four congruent sides and four $90°$ angles. d) Regular. A rectangle with equal length and width is a square, so all four sides are $6$ in and all four angles are $90°$.
7. a) regular octagon b) regular pentagon c) regular decagon
8. $P = 6 \times 12 = 72$ cm
9. $s = 60 \div 5 = 12$ in
10. $6 \times 120° = 720°$
11. $P = 8 \times 12 = 96$ in of tape. Since $96 \div 12 = 8$, that is $8$ ft.
12. A rhombus with four $8$ cm sides and angles of $70°$, $110°$, $70°$, $110°$ has all sides congruent but is not regular. It fails the angle test, because $70 \ne 110$. Congruent sides alone are not enough; the angles must match too.

Exit ticket 15.2

1. All sides congruent and all angles congruent.
2. A nonagon.
3. $P = 6 \times 9 = 54$ cm
4. Most rectangles have two different side lengths, so even though all four angles are $90°$, the side test fails. The one kind that is regular is a rectangle whose length equals its width — a square.

Lesson 15.3 — Lines of Symmetry

Guided practice

1. $3$
2. $4$
3. $5$
4. $8$ lines of symmetry. Since $8 \div 2 = 4$, there are $4$ vertex-to-vertex lines and $4$ midpoint-to-midpoint lines.
5. Folding along a line of symmetry lands one part exactly on the other, with no overhang and no gap. Matching exactly means every length and every angle in one part equals its partner in the other, which is what congruent means.

Independent practice

6. a) $7$ b) $9$ c) $10$ d) $20$
7. $3$ pass through two opposite vertices and $3$ pass through the midpoints of opposite sides, for $6$ in all.
8. No. A pentagon has an odd number of sides, so no vertex has a vertex directly opposite it — each vertex faces a side. Every line of symmetry therefore runs from a vertex to the midpoint of the opposite side. Vertex-to-vertex symmetry lines only happen when the number of sides is even.
9. $2$ lines: the line through the midpoints of the two $3$ cm sides and the line through the midpoints of the two $7$ cm sides. The diagonals do not count because folding a $3$ by $7$ rectangle along a diagonal leaves parts of each half sticking out past the other — the halves have the same area but do not match point for point.
10. $15$ sides. Since $15$ is odd, every line runs from a vertex to the midpoint of the opposite side, so none joins two vertices.
11. Folding along a line of symmetry makes the two parts land exactly on each other, so the parts have equal side lengths and equal angle measures — they are congruent. A regular hexagon has $6$ lines of symmetry, so $6$ different folds would work.
12. Take a parallelogram with sides $4$ cm and $9$ cm and angles of $60°$ and $120°$, which is neither a rectangle nor a rhombus. It has $0$ lines of symmetry. Folding along a diagonal puts a $4$ cm side on top of a $9$ cm side, so the halves do not match; they have the same area, but matching area is not the same as matching point for point.

Exit ticket 15.3

1. $4$
2. $10$
3. The two parts are congruent — same size and same shape, matching exactly when folded.
4. All the sides of a regular polygon are congruent and all its angles are congruent, so the figure looks the same lined up with any one of its sides or vertices. That gives one line of symmetry for each side, so an $n$-sided regular polygon has $n$ of them.

Lesson 15.4 — Congruent and Noncongruent Polygons

Guided practice

1. Congruent. All three pairs of corresponding sides are congruent and all three pairs of corresponding angles are congruent.
2. Congruent. Corresponding sides measure $3$ ft and $3$ ft, and $9$ ft and $9$ ft, and every angle in both is $90°$.
3. Noncongruent. The pair of corresponding longer sides fails: $9 \ne 10$.
4. $ST = 14$ cm
5. Both are squares, so all eight angles are $90°$ and the corresponding angles are congruent. But congruence also requires every pair of corresponding sides to be congruent, and the two squares have different side lengths, so that requirement fails. Same shape is not enough; the size must match too.

Independent practice

6. a) Congruent. All sides are $6$ cm and all angles are $60°$, so every corresponding pair matches. b) Noncongruent. The corresponding sides are $6$ cm and $7$ cm, and $6 \ne 7$. c) Congruent. All sides are $10$ in and all angles are $90°$. d) Noncongruent. All eight sides are $10$ in, but the square's angles are $90°$ while the rhombus has $80°$ and $100°$ angles, so corresponding angles do not match.
7. The side corresponding to $\overline{BC}$ is $\overline{XY}$. The angle corresponding to $\angle D$ is $\angle Z$.
8. $DE = 9$ m, $EF = 5$ m, $DF = 11$ m, $m\angle E = 100°$
9. Yes, congruent. Turning a figure changes its position and the direction it faces, but it does not change any side length or any angle measure. Since all corresponding sides and angles still match, the triangles are congruent.
10. Congruent. Regular means all sides congruent and all angles congruent, so a regular hexagon is completely determined by its side length. Both have $3$ cm sides, so all six pairs of corresponding sides are congruent, and every angle in both is $120°$.
11. Not congruent. Four sides match, but the fifth measures $5.2$ cm against the template's $5$ cm, a difference of $0.2$ cm. That side must be trimmed by $0.2$ cm — and the stamping tool corrected — before the plate is congruent to the template.
12. "Same shape" means the figures look alike, with corresponding angles congruent and sides in the same proportion; a $4$ cm square and a $10$ cm square have the same shape. "Congruent" means same shape and same size, so every pair of corresponding sides must be congruent too. The $4$ cm and $10$ cm squares are noncongruent, because $4 \ne 10$. Figures with the same shape but different sizes are called similar.

Exit ticket 15.4

1. Congruent.
2. $m\angle Z = 35°$
3. Yes, they have the same shape — both are squares with all $90°$ angles. No, they are not congruent, because $4 \ne 9$, so the corresponding sides are not congruent.
4. Every pair of corresponding sides must be congruent and every pair of corresponding angles must be congruent. If even one pair fails, the polygons are noncongruent.

Chapter 15 Review

Part A — Identifying regular polygons (6.MG.4a)

1. All sides congruent and all angles congruent, both at once.
2. a) Regular. Four congruent sides and four $90°$ angles. b) Not regular. The angles are all $90°$, but $5 \ne 11$, so the sides are not all congruent. c) Not regular. The sides are all $9$ in, but $60 \ne 120$, so the angles are not all congruent. d) Regular. Three congruent sides and three $60°$ angles.
3. a) regular hexagon b) regular octagon c) regular decagon
4. $P = 5 \times 11 = 55$ cm
5. $s = 56 \div 8 = 7$ in

Part B — Lines of symmetry in regular polygons (6.MG.4b)

6. a) $3$ b) $4$ c) $6$ d) $10$
7. Each of the $5$ lines runs from a vertex through the midpoint of the opposite side. None joins two vertices because $5$ is odd, so no vertex has a vertex directly across from it — every vertex faces a side.
8. Since $8 \div 2 = 4$: $4$ lines join opposite vertices, and $4$ lines join the midpoints of opposite sides.
9. They are congruent — same size and same shape, matching exactly when the figure is folded along the line.
10. $2$ lines, each through the midpoints of a pair of opposite sides. The diagonals are not lines of symmetry because folding a $4$ by $10$ rectangle along a diagonal does not make the halves land on each other; parts of each half stick out past the other.

Part C — Congruence of segments and angles (6.MG.4c)

11. $\overline{AB} \cong \overline{CD}$
12. $m\angle J = 118°$
13. No. Congruent angles have equal measures, and $61 \ne 16$.
14. $\cong$ goes between figures and $=$ goes between numbers. Example of $\cong$: $\overline{AB} \cong \overline{CD}$. Example of $=$: $AB = CD$.

Part D — Congruent and noncongruent polygons (6.MG.4d)

15. a) Congruent. Corresponding sides $6$ and $6$, $9$ and $9$; all angles $90°$. b) Noncongruent. One pair of corresponding sides is $9$ cm and $8$ cm, and $9 \ne 8$. c) Congruent. Regular hexagons with the same side length have all corresponding sides congruent and all angles $120°$. d) Noncongruent. All angles are $90°$, but $3 \ne 12$, so no pair of corresponding sides is congruent.
16. $RS = 12$ cm, $RT = 7$ cm, $m\angle S = 45°$
17. The side corresponding to $\overline{RS}$ is $\overline{VW}$. The angle corresponding to $\angle Q$ is $\angle U$.

Part E — Mixed application and reasoning

18. $P = 6 \times 4 = 24$ cm. It has $6$ lines of symmetry. Any two such tiles are congruent because regular fixes every measurement once the side length is known: all six sides are $4$ cm and all six angles are $120°$, so every pair of corresponding sides and angles matches.
19. Bracket A is congruent to the template, because its sides are $8$ in, $15$ in, $17$ in, matching all three. Bracket B is noncongruent: its longest side is $16$ in instead of $17$ in, a difference of $1$ in, so one pair of corresponding sides fails and the bracket will not match the template.
20. Counterexample: a square with $3$ cm sides and a square with $8$ cm sides. All eight angles measure $90°$, so every pair of corresponding angles is congruent, yet $3 \ne 8$, so the polygons are noncongruent. The error is that matching angles fix the shape but say nothing about the size; congruence requires the corresponding sides to be congruent as well. Figures with the same shape but different sizes are similar.

Workbook-only items

Page 2, sentences. Two segments are congruent when they have the same length. Two angles are congruent when they have the same measure. Congruent means same size and same shape.

Page 2, congruent or not. a) $\cong$ b) $\ncong$ c) $\cong$ d) $\ncong$ e) $\cong$ f) $\ncong$

Page 2, fill in. $PQ = 21$ mm; $m\angle D = 47°$; $m\angle R = 64°$

Page 3, the two $45°$ angles. Yes, they are congruent. Longer rays do not make a bigger angle, because an angle measures the amount of turn between two rays and the rays continue as far as you choose to draw them.

Page 3, reading the marks. Segments: $\overline{PQ}$ and $\overline{RS}$; $\overline{QR}$ and $\overline{SP}$. Angles: $\angle P$ and $\angle R$; $\angle Q$ and $\angle S$.

Page 3, $\cong$ or $=$. $\overline{AB} \cong \overline{CD}$; $AB = CD$; $\angle P \cong \angle Q$; $m\angle P = m\angle Q$.

Page 3, cut it in half. Ribbon: $12.5$ cm each. Board: $9$ cm each.

Page 5, definition. A regular polygon has all sides congruent and all angles congruent.

Page 5, name table. 3 triangle · 4 quadrilateral · 5 pentagon · 6 hexagon · 7 heptagon · 8 octagon · 9 nonagon · 10 decagon

Page 5, angle table. Equilateral triangle $60°$; square $90°$; regular pentagon $108°$; regular hexagon $120°$; regular octagon $135°$; regular decagon $144°$.

Page 6, which test fails. Rectangle $3$ by $7$: fails the sides test. Rhombus with $70°$ and $110°$ angles: fails the angles test.

Page 6, regular or not table. Equilateral triangle, $6$ cm sides — regular; all sides and all angles congruent. Rhombus, $8$ cm sides, $70°$ and $110°$ — not regular; angles not all congruent. Square, $3$ in sides — regular; both tests pass. Rectangle $6$ in by $6$ in — regular; equal length and width make it a square. Rectangle $4$ cm by $9$ cm — not regular; sides not all congruent.

Page 6, perimeter table. Hexagon, $12$ cm → $72$ cm. Octagon, $5$ cm → $40$ cm. Hexagon, $9$ cm → $54$ cm. Pentagon, perimeter $60$ in → side $12$ in. Decagon, perimeter $70$ m → side $7$ m. Pentagon, perimeter $45$ in → side $9$ in.

Page 6, counterexample. Any rhombus that is not a square works, for example one with four $8$ cm sides and angles of $70°$, $110°$, $70°$, $110°$. All sides are congruent, but the angles are not, so it is not regular.

Page 8, definition blank. The two parts are congruent.

Page 8, symmetry table. Equilateral triangle: $3$ sides, $3$ lines. Square: $4$, $4$. Regular pentagon: $5$, $5$. Regular hexagon: $6$, $6$. Regular heptagon: $7$, $7$. Regular octagon: $8$, $8$. Regular decagon: $10$, $10$. Regular $20$-gon: $20$, $20$.

Page 8, the pattern. A regular polygon has exactly as many lines of symmetry as it has sides.

Page 9, sort the cases. Odd: every line runs from a vertex to the midpoint of the opposite side. Even: half the lines join opposite vertices, and half join the midpoints of opposite sides.

Page 9, table.

Page 9, few or no lines. Rectangle $3$ cm by $7$ cm: $2$ lines. The diagonals do not count because folding along a diagonal leaves parts of each half sticking out past the other, so the halves do not match point for point. Parallelogram that is not a rectangle or rhombus: $0$. Scalene triangle: $0$.

Page 9, work backward. $15$ sides. The lines are all vertex to side midpoint, because $15$ is odd, so no vertex has a vertex directly opposite it.

Page 9, apply it. The fold makes the two parts land exactly on each other, so all matching sides and angles are equal, which is what congruent means. $6$ different folds would work.

Page 11, definition blanks. Every pair of corresponding sides is congruent AND every pair of corresponding angles is congruent.

Page 11, six facts. $\overline{AB} \cong \overline{DE}$; $\overline{BC} \cong \overline{EF}$; $\overline{AC} \cong \overline{DF}$; $\angle A \cong \angle D$; $\angle B \cong \angle E$; $\angle C \cong \angle F$.

Page 11, order matters. From $ABCD \cong WXYZ$: the side matching $\overline{BC}$ is $\overline{XY}$; the angle matching $\angle D$ is $\angle Z$. From $PQRS \cong TUVW$: the side matching $\overline{RS}$ is $\overline{VW}$; the angle matching $\angle Q$ is $\angle U$.

Page 11, measurements. $DE = 9$ m; $EF = 5$ m; $DF = 11$ m; $m\angle E = 100°$.

Page 12, analysis. All corresponding angles congruent? Yes. All corresponding sides congruent? No. Congruent or noncongruent? Noncongruent. Same shape, different size, is called similar.

Page 12, congruent or noncongruent table. Two equilateral triangles, $6$ cm sides — C; all sides and angles match. Equilateral triangles $6$ cm and $7$ cm — N; corresponding sides differ. Two squares, $10$ in sides — C; all sides $10$ in, all angles $90°$. Square $10$ in and rhombus $10$ in with $80°$ and $100°$ — N; sides match but angles do not. Two rectangles each $6$ cm by $9$ cm — C; all corresponding sides and angles match. Rectangles $6$ by $9$ and $6$ by $8$ — N; $9 \ne 8$. Two regular hexagons, $5$ mm sides — C; regular plus equal side length forces every part to match. Squares $3$ in and $12$ in — N; $3 \ne 12$.

Page 13, item 1. Congruent. Turning changes the figure's position and the direction it faces. It does not change any side length or any angle measure.

Page 13, item 2. Congruent. Regular means all sides congruent and all angles congruent, so the side length alone determines the whole hexagon; both have $3$ cm sides and $120°$ angles.

Page 13, item 3. Not congruent. The $5$ cm side is wrong: it came out $5.2$ cm, which is $0.2$ cm too long. That side must be trimmed by $0.2$ cm, and the stamping tool corrected, so the plate matches the template.

Page 13, item 4. Bracket A is congruent. Bracket B has a $16$ in side where the template has $17$ in, a difference of $1$ in, so one pair of corresponding sides is not congruent and the bracket is noncongruent.

Page 13, item 5. A square with $3$ cm sides and a square with $8$ cm sides have all corresponding angles congruent at $90°$ but are noncongruent, because $3 \ne 8$. The correct word for those figures is similar.

Pages 15–16, Chapter 15 review. Same items as the textbook review; see the Chapter 15 Review key above.