Congruence proofs: fill in the missing step or reason

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Use this

   Given: Q is the midpoint of DG and of FH. Prove: DH ≅ GF. Ladder: 1. Q is the midpoint of DG -- Given; 2. DQ ≅ GQ -- Definition of midpoint; 3. Q is the midpoint of FH -- Given; 4. HQ ≅ FQ -- Definition of midpoint; 5. ∠DQH ≅ ∠GQF -- Vertical angles theorem; 6. triangle DQH ≅ triangle GQF -- SAS; 7. DH ≅ GF -- ?.

2. 1.

   Ren opens scissors so the blades DG and FH cross at the pivot Q. Which reason fills the blank?

   • Given
   • CPCTC
   • SAS
   • Definition of midpoint

3. Use this

   Given: C is the midpoint of SV and of UW. Prove: SW ≅ VU. Ladder: 1. C is the midpoint of SV -- Given; 2. SC ≅ VC -- Definition of midpoint; 3. C is the midpoint of UW -- Given; 4. WC ≅ UC -- Definition of midpoint; 5. ∠SCW ≅ ∠VCU -- Vertical angles theorem; 6. triangle SCW ≅ triangle VCU -- SAS; 7. SW ≅ VU -- ?.

4. 2.

   Tala crosses two craft sticks SV and UW at their middles, C. Which reason fills the blank?

   • CPCTC
   • Vertical angles theorem
   • SAS
   • Given

5. Use this

   Sketch, not to scale: segments DG and FH cross at Q (D upper left, G lower right, H upper right, F lower left). Marked: arrows on DH and FG; DQ and QG with one tick each. Given: DH ∥ FG, and Q is the midpoint of DG. Prove: HQ ≅ FQ. Ladder: 1. DH ∥ FG -- Given; 2. ∠HDQ ≅ ∠FGQ -- Alternate interior angles theorem; 3. Q is the midpoint of DG -- Given; 4. DQ ≅ GQ -- Definition of midpoint; 5. ∠DQH ≅ ∠GQF -- Vertical angles theorem; 6. triangle DQH ≅ triangle GQF -- ?; 7. HQ ≅ FQ -- CPCTC.

6. 3.

   Zee draws a bike frame where bars DG and FH cross at Q, and DH ∥ FG. Which reason fills the blank?

   • AAS
   • ASA
   • SAS
   • SSS

7. Use this

   Ladder: 1. SW ∥ UV -- Given; 2. ∠WSC ≅ ∠UVC -- Alternate interior angles theorem; 3. SW ≅ VU -- Given; 4. ∠SCW ≅ ∠VCU -- Vertical angles theorem; 5. triangle SCW ≅ triangle VCU -- ?; 6. WC ≅ UC -- CPCTC.

8. 4.

   Ari draws two ramps that cross, SV and UW, meeting at C, with SW ∥ UV. Which reason fills the blank?

   • AAS
   • SAS
   • HL
   • ASA

9. Use this

   triangles CDF and CGF share side CF; D is above CF and G is below, with right angles at D and G. Marked: right-angle boxes at D and G; CD and CG with one tick each. Given: ∠CDF and ∠CGF are right angles, and CD ≅ CG. Prove: DF ≅ GF. Ladder: 1. ∠CDF and ∠CGF are right angles -- Given; 2. triangle CDF and triangle CGF are right triangles -- Definition of right triangle; 3. CD ≅ CG -- Given; 4. CF ≅ CF -- Reflexive property; 5. triangle CDF ≅ triangle CGF -- ?; 6. DF ≅ GF -- CPCTC.

10. 5.

    Bo builds a kite from two right triangles, CDF and CGF, that share stick CF. Which reason fills the blank?

    • HL
    • SSA
    • SAS
    • ASA

11. Use this

    Given: ∠SUW and ∠SYW are right angles, and SU ≅ SY. Prove: UW ≅ YW. Ladder: 1. ∠SUW and ∠SYW are right angles -- Given; 2. triangle SUW and triangle SYW are right triangles -- Definition of right triangle; 3. SU ≅ SY -- Given; 4. SW ≅ SW -- Reflexive property; 5. triangle SUW ≅ triangle SYW -- ?; 6. UW ≅ YW -- CPCTC.

12. 6.

    Esme folds paper into two right triangles, SUW and SYW, that share fold SW. Which reason fills the blank?

    • SAS
    • SSA
    • HL
    • AAS
13. 7.

    Jun opens garden shears so the blades HU and YD cross at their middles, F. Which statement fills the blank?

    • △HFD ≅ △UFY
    • △HFD ≅ △YFU
    • △HFD ≅ △HFY
    • HD ≅ UY

14. Use this

    Given: HD ∥ YU, and F is the midpoint of HU. Prove: DF ≅ YF. Ladder: 1. HD ∥ YU -- Given; 2. ? -- Alternate interior angles theorem; 3. F is the midpoint of HU -- Given; 4. HF ≅ UF -- Definition of midpoint; 5. ∠HFD ≅ ∠UFY -- Vertical angles theorem; 6. triangle HFD ≅ triangle UFY -- ASA; 7. DF ≅ YF -- CPCTC.

15. 8.

    Kira draws a bike rack where bars HU and YD cross at F, and HD ∥ YU. Which statement fills the blank?

    • ∠DHF ≅ ∠YUF
    • ∠DHF ≅ ∠HDF
    • ∠DHF ≅ ∠UYF
    • DF ≅ YF
16. 9.

    Luz sews two right-triangle patches, QHV and QDV, that share seam QV. Which statement fills the blank?

    • HV ≅ DV
    • △QHV and △QDV are right triangles
    • QV ⊥ HD
    • △QHV ≅ △QDV

17. Use this

    Given: ∠GCS and ∠GWS are right angles, and GC ≅ GW. Prove: CS ≅ WS. Ladder: 1. ∠GCS and ∠GWS are right angles -- Given; 2. triangle GCS and triangle GWS are right triangles -- Definition of right triangle; 3. GC ≅ GW -- Given; 4. ? -- Reflexive property; 5. triangle GCS ≅ triangle GWS -- HL; 6. CS ≅ WS -- CPCTC.

18. 10.

    Milo cuts a sandwich into two right triangles, GCS and GWS, along the shared cut GS. Which statement fills the blank?

    • GC ≅ CS
    • GC ≅ GS
    • GS ≅ GS
    • CS ≅ WS