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?

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

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

   • 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
   • Yes: any three equal parts settle that two triangles are congruent

3. Read this first

   Use this

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

4. 3.

   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

5. Statement

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

6. 4.

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

   • 130°
   • 40°
   • 50°
   • 90°
7. 5.

   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 area 24, so they must be congruent
   • 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 a right angle, so right angles settle congruence
8. 6.

   Kira's scooter dot P(2, 1) is flipped across the line x = 0, then across the line x = 3. Which single rigid motion does that pair equal?

   • A slide 6 units left
   • A slide 6 units right, twice the gap between the lines
   • A flip across the line x = 1.5
   • A slide 3 units right, the same as the distance between the lines

9. Use this

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

10. 7.

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

    • ASA: AB lies between the equal angles, so the rays from its ends meet alike
    • AAA: the three angles are equal, so the two triangles must be congruent
    • No criterion fits: two angles and one side are not enough to settle congruence
    • AAS: the equal side is not between the equal angles, so ASA cannot be used
11. 8.

    A counterexample must have all four sides equal in each quadrilateral, but the shapes are not congruent. Noah says if all four sides of two quadrilaterals are equal in length, the quadrilaterals are congruent. Which pair breaks that claim?

    • Two squares with side 6, one turned
    • A kite and a rhombus, both with the long side equal to 6
    • A square and a rectangle with the same perimeter of 24
    • A square and a leaning rhombus, both with side 6

12. 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.

13. 9.

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

    • △MNP ≅ △YXZ
    • △MNP ≅ △ZYX
    • △MNP ≅ △XYZ
    • △MNP ≅ △YZX
14. 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?

    • 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
    • A proof must use six parts of the triangles, and this one does not
    • It assumes the congruence to get AB = DE, which is what the proof must show