Geometry Workbook — Chapter 6: Congruence by Algebra and Coordinates

SOL G.TR.2 (b, c, e) · Companion to Textbook Chapter 6

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/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 88.

PAGE 1 — Chapter opener

Chapter 6 · Congruence by Algebra and Coordinates

Standard G.TR.2 (b, c, e)

In this chapter you will:

• Turn a pair of tick marks into an equation, and report the length rather than the variable
• Check twice — substitute back, then confirm a triangle can exist
• Use distance, slope, and midpoint on lattice points, in exact form
• Prove congruence from coordinates alone, by SSS and by SAS with a right included angle
• Spot a pair that is congruent but written in the wrong order
• Read measured attributes off a congruence you proved

Words to know: algebraic method · coordinate method · distance formula · slope · negative reciprocal · perpendicular · midpoint · lattice point · simplest radical form · correspondence · CPCTC · perimeter · Triangle Inequality

Conventions: a variable is not a length — solve, then substitute back. Exact before approximate — $2\sqrt{5}$, not $4.47$. Slope certifies exactly one angle measure: $90°$.

PAGE 2 — Marks become equations

6.1 A Tick Mark Is an Equation

FIGURE: fig1-marks-become-equations.png (full width)

Fill in the blanks.

Congruent segments have equal ____________ , so a pair of matching tick marks can be written as an ____________ .

Three pairs of marks give ______ equations, each with ______ variable.

1. Which pair of sides do the single tick marks name? ______

2. Write the equation the double tick marks force. ____________________

3. Solve $2w + 7 = 5w - 11$. $w =$ ______ $AC =$ ______

PAGE 3 — Solve, then check twice

The Two Checks

FIGURE: fig2-solve-then-check.png (full width)

Fill in the blanks.

Check 1: substitute the value back into ______ expressions and confirm they ____________ .

Check 2: confirm the three lengths satisfy the ____________________ .

4. Why is $x = 5$ not the length of $\overline{AB}$?

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5. Show the Triangle Inequality check for the solved sides $17$, $23$, $19$.

   ______ + ______ = ______ > ______ ✓

PAGE 4 — Angles give equations too

Algebraic Angle Measures

FIGURE: fig3-algebraic-angles-and-asa.png (full width)

Fill in the blanks.

Congruent angles have equal ____________ , so a pair of matching arcs is also an ____________ .

Solving the algebra does not name the criterion. The ____________________ does.

6. Which criterion do the solved parts land on, and why is it not AAS?

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PAGE 5 — Practice · algebraic congruence

Practice

7. $AB = 2x + 9$, $DE = 6x - 7$. $x =$ ______ $AB =$ ______ $DE =$ ______

8. $BC = 7y - 4$, $EF = 3y + 20$. $y =$ ______ $BC =$ ______ $EF =$ ______

9. $m\angle A = (5a - 12)°$, $m\angle D = (3a + 8)°$. $a =$ ______ $m\angle A =$ ______ $m\angle D =$ ______

10. $m\angle B = (2b + 31)°$, $m\angle E = (7b - 44)°$. $b =$ ______ $m\angle B =$ ______ $m\angle E =$ ______

11. $\triangle ABC \cong \triangle DEF$: $AB = 4x + 1$, $DE = 2x + 13$; $BC = 3y - 2$, $EF = y + 12$; $AC = 5z - 8$, $DF = 3z + 4$.

    $DE =$ ______ $EF =$ ______ $DF =$ ______ Perimeter of $\triangle DEF =$ ______

12. Show the three lengths from item 11 can form a triangle. ______ + ______ = ______ > ______

13. $m\angle A = (3a + 5)°$, $m\angle D = (a + 41)°$; $m\angle B = (6b - 7)°$, $m\angle E = (2b + 45)°$.

    Angles of $\triangle DEF$: ______ , ______ , ______

14. $AB = 2x + 3$, $DE = 2x + 9$, marked congruent. What does solving tell you about the figure?

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15. Solved sides $8$, $5$, $14$. Triangle? ______ Inequality: ____________________

16. Two angles of one triangle solve to $96°$ and $88°$. What is wrong? ____________________

17. Solve $6x - 5 = 2x + 27$. $x =$ ______ Length = ______

18. Solve $(2a + 18)° = (5a - 21)°$. $a =$ ______ Measure = ______

19. Solved sides $9$, $12$, $22$. Possible? ______ Check: ____________________

PAGE 6 — Apply and reason · Lesson 6.1

Think It Through

14. Error analysis. A student solves $3x + 2 = 5x - 8$ and writes "$AB = 5$." Error and correction:

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15. Application. A truss template's long edge is $(4x + 6)$ in; the cut piece's corresponding edge is $(7x - 9)$ in.

    $x =$ ______ Long edge = ______ in

16. Reasoning. Why is an algebraic congruence problem not finished when the variable is found? Name both remaining steps.

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17. Error analysis. Given $\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$, $\angle C \cong \angle F$, a student names ASA. Correct them.

    Criterion that actually applies: ______ Why: ____________________

18. Name the two checks that finish every algebraic congruence problem.

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PAGE 7 — Distance

6.2 Distance Is the Pythagorean Theorem

FIGURE: fig4-distance-from-the-pythagorean-theorem.png (full width)

Fill in the blanks.

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The two differences are the ______ of a right triangle; the segment is the ____________ .

Squaring removes the ______ , so the order of subtraction does not matter.

25. Leg lengths: ______ and ______ From which subtractions? ____________________

26. $AB^2 =$ ______ + ______ = ______ $AB =$ ______

PAGE 8 — Slope

Slope Certifies One Angle

FIGURE: fig5-slope-decides-a-right-angle.png (full width)

Fill in the blanks.

Equal slopes mean the segments are ____________ .

A slope product of $-1$ means the segments are ______________ , so the angle is ______ .

A vertical segment's slope is ____________ .

27. Slope of $\overline{BA}$ = ______ Slope of $\overline{BC}$ = ______ Product = ______

28. State exactly what that product proves. ____________________

PAGE 9 — Midpoint

Midpoint, and the One Half Unit

FIGURE: fig6-midpoint-and-a-half-unit.png (full width)

Fill in the blanks.

$$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$$

Working backwards from $A$ and $M$: $B = ($ ______ , ______ $)$.

29. Midpoint of the segment from $(-5, -3)$ to $(3, 5)$: ______

30. Why is the second midpoint not a lattice point? ____________________

PAGE 10 — Practice · the three tools

Practice

31. $(2, -3)$ to $(7, 9)$: $d =$ ______

32. $(-4, 1)$ to $(2, 9)$: $d =$ ______

33. $(-1, -2)$ to $(3, 1)$: $d =$ ______

34. $(0, 5)$ to $(6, -1)$: $d =$ ______ (simplest radical form)

35. $(-3, 4)$ to $(2, -1)$: $d =$ ______ (simplest radical form)

36. Slope from $(-6, 2)$ to $(2, 6)$: ______

37. Slope from $(5, -1)$ to $(5, 7)$: ______

38. Perpendicular? $(-4, 0)$ to $(0, 3)$, and $(0, 3)$ to $(6, -5)$.

    Slopes ______ and ______ Product ______ Perpendicular? ______

39. Midpoint of $(-7, 2)$ and $(3, -6)$: ______

40. Midpoint of $(-4, 1)$ and $(3, 6)$: ______

41. $M(1, 4)$ is the midpoint of $\overline{AB}$, $A(-3, -2)$. $B =$ ______

42. $(-2, -5)$ to $(4, 3)$: $d =$ ______

43. Slope from $(-1, 6)$ to $(5, 2)$: ______

44. Midpoint of $(-9, 4)$ and $(2, -3)$: ______

PAGE 11 — Apply and reason · Lesson 6.2

Think It Through

42. Application. A drone flies from $(-5, -2)$ to $(7, 3)$ on a map whose grid units are $100$ m.

    Distance in units = ______ Distance in metres = ______

43. Reasoning. A student says the distance from $(2, 5)$ to $(8, 13)$ is $6 + 8 = 14$. Explain the error and give the correct distance.

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    Correct distance = ______

44. Which of the three tools certifies a right angle, and what exactly must it show?

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PAGE 12 — SSS on the plane

6.3 SSS Is the Workhorse

FIGURE: fig7-sss-on-the-coordinate-plane.png (full width)

The four steps.

1. Plot and name the ____________________ you intend to prove.

2. Compute the parts the criterion needs, in ____________ form.

3. Compare them in ____________ order.

4. Name the ____________ and state the congruence.

5. $AB =$ ______ $DE =$ ______

6. $BC =$ ______ $EF =$ ______

7. Criterion ______ Statement ____________________

PAGE 13 — SAS with a right included angle

When SAS Is Available

FIGURE: fig8-sas-with-a-right-included-angle.png (full width)

Fill in the blanks.

Coordinates give lengths exactly and give exactly one angle measure: ______ , certified by a slope product of ______ .

For any other included angle, compute the third ____________ and use ______ instead.

51. Slope of $\overline{PQ}$ = ______ Slope of $\overline{PR}$ = ______ What that proves: ____________________

52. Criterion ______ The three parts it used: ____________________

PAGE 14 — Congruent, wrong order

The Order Is Part of the Claim

FIGURE: fig9-congruent-but-not-in-that-order.png (full width)

Fill in the blanks.

Compute all ______ distances first, then let them tell you the ____________ .

53. Which comparison fails first? ______ Why does that not mean the triangles are different?

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    Correspondence that works: ____________________

PAGE 15 — Practice · coordinate proofs

Practice

54. $A(-5, 4)$, $B(-1, 5)$, $C(-3, 1)$; $D(2, -4)$, $E(6, -3)$, $F(4, -7)$.

    $AB$ ______ $BC$ ______ $AC$ ______ $DE$ ______ $EF$ ______ $DF$ ______ Criterion ______

55. $P(-6, 1)$, $Q(-2, 3)$, $R(-4, -2)$; $S(1, -3)$, $T(3, -7)$, $U(-2, -5)$.

    $PQ$ ______ $QR$ ______ $PR$ ______ $ST$ ______ $TU$ ______ $SU$ ______ Congruent? ______

56. $A(0, 0)$, $B(6, 0)$, $C(0, 4)$; $D(-7, 1)$, $E(-1, 1)$, $F(-7, 6)$.

    Congruent? ______ One comparison that settles it: ____________________

57. $A(-6, -2)$, $B(-1, -2)$, $C(-4, 2)$; $D(1, -2)$, $E(6, -2)$, $F(4, 2)$.

    Comparison that fails: ____________________ True statement: ____________________

58. $A(-5, -3)$, $B(-1, -1)$, $C(-6, -1)$; $D(4, -1)$, $E(6, -5)$, $F(6, 0)$.

    $AB$ ______ $AC$ ______ Slopes at $A$ ______ and ______ $DE$ ______ $DF$ ______ Slopes at $D$ ______ and ______

    Criterion ______

59. $A(-2, 1)$, $B(2, 3)$, $C(4, -1)$. Show $\angle B$ is right two ways.

    Slopes at $B$: ______ and ______ Product ______

    Squared sides: ______ + ______ = ______

60. $A(0, 0)$, $B(4, 2)$, $C(1, 5)$; $D(6, 1)$, $E(10, 3)$, $F(7, 6)$. $AB =$ ______ $DE =$ ______

    Does that pair alone prove anything? ______ Why? ____________________

61. $(-3, -1)$ to $(1, 1)$ and $(1, 1)$ to $(3, -3)$: slopes ______ and ______ Product ______ Right angle? ______

62. A proof verifies $AB = DE$, $BC = EF$, $AC = DF$. Criterion: ______

PAGE 16 — Apply and reason · Lesson 6.3

Think It Through

59. Reasoning. Why is SSS, rather than ASA or AAS, the criterion nearly every coordinate proof uses?

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60. Error analysis. A student gets $\sqrt{193}$ and $\sqrt{194}$, rounds both to $13.9$, and declares congruence. Why does the conclusion not follow?

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61. Application. Flower beds at $(-6, 1)$, $(-2, 3)$, $(-4, -2)$ and $(1, -3)$, $(3, -7)$, $(-2, -5)$; one grid unit is one metre. Will one edging kit fit both?

    Answer ______ Reason ____________________ Perimeter of each ≈ ______ m

62. Reasoning. Why is SAS available on the coordinate plane only when the included angle is right, and what do you do instead?

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63. Error analysis. After verifying $AB = DE$, $BC = EF$, $AC = DF$, a student writes $\triangle ABC \cong \triangle DFE$. Error:

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64. State the four steps of a coordinate proof, in order.

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PAGE 17 — Work frames

Blank Frames

FIGURE: fig12-blank-grids-and-proof-frames.png (full width)

Use the empty plane to plot any pair of triangles a problem gives you, the six-length table to record the distances before you compare them, and the five-row frame to write the proof.

Reminder. Fill the length table completely before writing a correspondence. A comparison that fails tells you the order is wrong; it does not tell you the triangles are different.

PAGE 18 — Attributes from a proved congruence

6.4 Read the Attributes Off the Proof

FIGURE: fig10-attributes-from-coordinates.png (full width)

The four-step method.

1. Prove the ____________________ .

2. Write the ____________________ in order.

3. Match the part you ______ to the part you ______ .

4. Name ____________ as the reason.

5. $AB =$ ______ $BC =$ ______ $AC =$ ______

6. Perimeter of $\triangle DEF =$ ______ Why no sides of $\triangle DEF$ were measured: ____________________

7. $m\angle E =$ ______ Reason: ____________

PAGE 19 — In context

Across the Creek

FIGURE: fig11-coordinate-proof-in-context.png (full width)

Fill in the blanks.

Convert to real units ______ , at the end — work in grid units first.

An answer is half an answer without its ____________ .

72. One grid unit = ______ m. Near plot's sides: ______ m, ______ m, ______ m

73. Fence needed by the far plot: ______ m

74. Criterion that licensed item 73: ______

PAGE 20 — Practice · measured attributes

Practice

75. $\triangle ABC \cong \triangle DEF$, $AB = 2\sqrt{13}$, $BC = 5$, $AC = 7$.

    Perimeter of $\triangle DEF$ (exact) = ______ (nearest hundredth) ≈ ______

76. $A(-3, -2)$, $B(5, -2)$, $C(5, 4)$. $AB$ ______ $BC$ ______ $AC$ ______ Perimeter ______

77. $\triangle PQR \cong \triangle STU$, $PQ = 3x + 4$, $ST = 5x - 10$. $x =$ ______ $ST =$ ______

78. $\triangle ABC \cong \triangle DEF$, $m\angle A = 47°$, $m\angle B = 68°$. $m\angle D$ ______ $m\angle E$ ______ $m\angle F$ ______

79. Vertices $(-1, -3)$, $(3, 0)$, $(-1, 0)$. Perimeter ______ Right angle at ______

80. $\triangle ABC \cong \triangle DEF$, perimeter of $\triangle ABC = 34$, $DE = 11$, $EF = 9$. $DF =$ ______ $AC =$ ______

81. $\triangle ABC \cong \triangle DEF$, $AB = 9$, $BC = 12$, $AC = 15$. Perimeter of $\triangle DEF =$ ______

82. $A(2, -1)$, $B(2, 5)$, $C(10, -1)$. Perimeter = ______

83. $\triangle ABC \cong \triangle DEF$, $m\angle C = 39°$. Angle of $\triangle DEF$ measuring $39°$: ______ Reason: ______

PAGE 21 — Apply and reason · Lesson 6.4

Apply It

81. Application. A bracket template has corners at $(0, 0)$, $(12, 0)$, $(0, 5)$ on a table marked in centimetres.

    Sides ______ , ______ , ______ Perimeter of every bracket ______ cm Reason ____________

82. Application. Two congruent sails, sides $6$, $8$, $10$ m; binding costs $14 per metre.

    Perimeter of one ______ m Total binding ______ m Cost ______

83. Error analysis. $\triangle ABC \cong \triangle DEF$, $AC = 15$; a student reports $DE = 15$. Correction:

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84. Reasoning. Why does proving a congruence let you report an attribute you never measured? Name the reason you write beside that step.

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85. State the four-step method for finding a measured attribute of a triangle you cannot reach.

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PAGE 22 — Chapter review

Review

Review 1 (G.TR.2b). $AB = 5x - 6$, $DE = 3x + 10$; $m\angle B = (4y + 3)°$, $m\angle E = (6y - 25)°$; $BC = 2z + 11$, $EF = 4z - 7$.

• $AB =$ ______ $m\angle B =$ ______ $BC =$ ______ Substitution check for one pair: ____________________
• $\angle B$ sits between $\overline{AB}$ and $\overline{BC}$. Criterion: ______
• A classmate answers "$x = 8$, $y = 14$, $z = 9$" and stops. What is still missing? ____________________
• If the third pair had been $AC = 2z + 11$ and $DF = 4z - 7$: criterion ______ What changed: ____________________

Review 2 (G.TR.2c). $A(-7, 1)$, $B(-3, 4)$, $C(-2, -2)$; $D(4, -3)$, $E(1, 1)$, $F(7, 2)$.

• $AB$ ______ $BC$ ______ $AC$ ______ $DE$ ______ $EF$ ______ $DF$ ______
• Statement ____________________ Criterion ______
• Why no angle measure was needed: ____________________
• A classmate writes $\triangle ABC \cong \triangle EDF$. One comparison that shows it false: ____________________ Why the triangles are still congruent: ____________________

Review 3 (G.TR.2 c, e). One grid unit is $4$ metres. Bed 1: $(-8, -1)$, $(-2, -1)$, $(-8, 7)$. Bed 2: $(2, 6)$, $(2, 0)$, $(10, 6)$.

• Criterion ______ Correspondence ____________________
• Bed 1's sides in metres: ______ , ______ , ______
• Edging for Bed 2: ______ m Reason: ____________
• At $7.50 per metre, cost for Bed 2: ______ Why no measurement across the wall is needed: ____________________

Canva production notes

• Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
• Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
• Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white.
• Coordinate work needs room. Items 54–58, 60, 65, and Reviews 2 and 3 each ask for six distances. Give every one of those items at least six ruled blanks on one line, or a small three-by-two grid of blanks, so students are not tempted to skip lengths.
• Radicals must render. The blanks on pages 10, 15, and 20 will be filled with answers like $2\sqrt{5}$ and $\sqrt{37}$. Leave the blanks tall enough for a radical sign with an overbar, not just a digit.
• The blank plane on page 17 is fig12-blank-grids-and-proof-frames.png. Reprint it for any extra coordinate problem a teacher assigns; it is the only page students are expected to photocopy.
• Symbols: ≅, ≇, ∠, △, ⊥, ∥, √, ↔, and the overline for segments must render in the body font; check the Canva font supports them before the first export.
• Item numbers are continuous from 1 to 88 and must not be renumbered when pages are reordered.