Congruent figures: corresponding parts and rigid motions

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   A matches D, B matches E, C matches F. Luz's quilt piece △ABC has A(1, 2), B(4, 2), C(4, 4). Its image is D(-4, 2), E(-7, 2), F(-7, 4). Which sequence of moves maps △ABC onto △DEF?

   • Turn 180° about the origin, then slide 3 left
   • Slide 3 right, then flip across the y-axis
   • Flip across the y-axis, then slide 3 right
   • Flip across the x-axis, then slide 3 right
2. 2.

   Ari's kite sail △ABC has A(3, 1), B(6, 1), C(6, 4). It turns 90° counterclockwise about A. Where does C land?

   • C′(-3, 3)
   • C′(6, -2)
   • C′(0, -2)
   • C′(0, 4)
3. 3.

   In two triangles, AB = DE, BC = EF and ∠A = ∠D. Tala says SAS proves them congruent. Is Tala right?

   • Yes: any three equal parts settle that two triangles are congruent
   • No: congruence needs the six parts shown equal
   • Yes: two sides and an angle are equal, so this is SAS and the triangles match
   • No: ∠A is not between the equal sides, so the parts can fit two different triangles

4. Read this first

   Use this

   AB = DE, ∠A = ∠D, AC = DF. After sliding A onto D, the triangles share only the corner D.

5. 4.

   Ren shows SAS works. Triangles have AB = DE, ∠A = ∠D and AC = DF. A is slid onto D. Which moves finish the argument?

   • Turn about D until C reaches F, which works for any two triangles that have AB = DE
   • Because ∠A = ∠D, the third sides already match, so no more moves are needed
   • Turn so AB lies along DE, flip if needed so C is on F's side, then AC = DF puts C on F
   • Enlarge until AC = DF, since the triangles need the same size to fit

6. Statement

   △ABC ≅ △DEF. ∠A = (3x + 5)°, ∠D = (5x − 25)°, ∠B = (2x + 10)°. ∠F is not given.

7. 5.

   Esme's paper triangles satisfy △ABC ≅ △DEF. ∠A = (3x + 5)°, ∠D = (5x − 25)° and ∠B = (2x + 10)°. How big is ∠F?

   • 40°
   • 130°
   • 50°
   • 90°
8. 6.

   Triangle 1 has corners (0, 0), (6, 0), (0, 8). Triangle 2 has (1, 1), (1, 7), (9, 1). Zee says they are congruent. Which reason is best?

   • Both have sides 6, 8 and 10, so SSS shows a rigid motion fits one onto the other
   • Their corners sit at different points on the grid, so the two cannot be congruent
   • Both have area 24, so they must be congruent
   • Both have a right angle, so right angles settle congruence

9. Statement

   △JKL ≅ △RST. JK = 2x + 7, RS = 4x − 5, KL = 3y − 1, ST = y + 9, in centimeters.

10. 7.

    Bo's sails satisfy △JKL ≅ △RST. JK = 2x + 7, RS = 4x − 5, KL = 3y − 1 and ST = y + 9. How long is KL?

    • 28 cm
    • 5 cm
    • 14 cm
    • 19 cm

11. Use this

    ∠A = ∠D = 40°, ∠B = ∠E = 65°, AB = DE = 7. Side AB lies between ∠A and ∠B.

12. 8.

    Pia studies △ABC and △DEF with ∠A = ∠D = 40°, ∠B = ∠E = 65° and AB = DE = 7. Which criterion proves them congruent, and why?

    • AAA: the three angles are equal, so the two triangles must be congruent
    • ASA: AB lies between the equal angles, so the rays from its ends meet alike
    • AAS: the equal side is not between the equal angles, so ASA cannot be used
    • No criterion fits: two angles and one side are not enough to settle congruence

13. Use this

    △MNP: ∠M = 38°, ∠N = 72°, MN = 9 cm. △XYZ: ∠Y = 38°, ∠Z = 72°, YZ = 9 cm. Each 9 cm side lies between its two given angles.

14. 9.

    Milo compares △MNP and △XYZ. ∠M = 38°, ∠N = 72°, MN = 9; ∠Y = 38°, ∠Z = 72°, YZ = 9. Which statement is correct?

    • △MNP ≅ △XYZ
    • △MNP ≅ △YXZ
    • △MNP ≅ △YZX
    • △MNP ≅ △ZYX
15. 10.

    The proof uses AB = DE as a reason. The problem gave only BC = EF and ∠B = ∠E. Jun writes: "The triangles are congruent, so AB = DE. Also BC = EF and ∠B = ∠E, so SAS makes them congruent." What is wrong?

    • It assumes the congruence to get AB = DE, which is what the proof must show
    • A proof must use six parts of the triangles, and this one does not
    • SAS needs two angles and a side, but the proof has two sides and an angle left
    • AB = DE is true because congruent triangles have equal sides, so the proof is fine