Congruent figures: corresponding parts and rigid motions
Grade 10 Math Worksheet
Circle the best answer.
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′(6, -2)
- C′(0, -2)
- C′(-3, 3)
- C′(0, 4)
In two triangles, AB = DE, BC = EF and ∠A = ∠D. Tala says SAS proves them congruent. Is Tala right?
- No: ∠A is not between the equal sides, so the parts can fit two different triangles
- Yes: two sides and an angle are equal, so this is SAS and the triangles match
- No: congruence needs the six parts shown equal
- Yes: any three equal parts settle that two triangles are congruent
Read this first
Use this
AB = DE, ∠A = ∠D, AC = DF. After sliding A onto D, the triangles share only the corner D.
Ren shows SAS works. Triangles have AB = DE, ∠A = ∠D and AC = DF. A is slid onto D. Which moves finish the argument?
- 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
- Because ∠A = ∠D, the third sides already match, so no more moves are needed
- Turn about D until C reaches F, which works for any two triangles that have AB = DE
Statement
△ABC ≅ △DEF. ∠A = (3x + 5)°, ∠D = (5x − 25)°, ∠B = (2x + 10)°. ∠F is not given.
Esme's paper triangles satisfy △ABC ≅ △DEF. ∠A = (3x + 5)°, ∠D = (5x − 25)° and ∠B = (2x + 10)°. How big is ∠F?
- 40°
- 90°
- 130°
- 50°
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?
- 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
- Both have sides 6, 8 and 10, so SSS shows a rigid motion fits one onto the other
- Both have area 24, so they must be congruent
Statement
△JKL ≅ △RST. JK = 2x + 7, RS = 4x − 5, KL = 3y − 1, ST = y + 9, in centimeters.
Bo's sails satisfy △JKL ≅ △RST. JK = 2x + 7, RS = 4x − 5, KL = 3y − 1 and ST = y + 9. How long is KL?
- 14 cm
- 5 cm
- 19 cm
- 28 cm
Use this
∠A = ∠D = 40°, ∠B = ∠E = 65°, AB = DE = 7. Side AB lies between ∠A and ∠B.
Pia studies △ABC and △DEF with ∠A = ∠D = 40°, ∠B = ∠E = 65° and AB = DE = 7. Which criterion proves them congruent, and why?
- 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
- 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
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?
- A square and a rectangle with the same perimeter of 24
- Two squares with side 6, one turned
- A kite and a rhombus, both with the long side equal to 6
- A square and a leaning rhombus, both with side 6
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.
Milo compares △MNP and △XYZ. ∠M = 38°, ∠N = 72°, MN = 9; ∠Y = 38°, ∠Z = 72°, YZ = 9. Which statement is correct?
- △MNP ≅ △XYZ
- △MNP ≅ △ZYX
- △MNP ≅ △YZX
- △MNP ≅ △YXZ
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?
- A proof must use six parts of the triangles, and this one does not
- AB = DE is true because congruent triangles have equal sides, so the proof is fine
- It assumes the congruence to get AB = DE, which is what the proof must show
- SAS needs two angles and a side, but the proof has two sides and an angle left