Congruence proofs: fill in the missing step or reason
Grade 10 Math Worksheet
Circle the best answer.
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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 -- ?.
Ren opens scissors so the blades DG and FH cross at the pivot Q. Which reason fills the blank?
- SAS
- CPCTC
- Given
- Definition of midpoint
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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 -- ?.
Tala crosses two craft sticks SV and UW at their middles, C. Which reason fills the blank?
- Given
- Vertical angles theorem
- CPCTC
- SAS
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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.
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
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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.
Ari draws two ramps that cross, SV and UW, meeting at C, with SW ∥ UV. Which reason fills the blank?
- ASA
- HL
- AAS
- SAS
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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.
Esme folds paper into two right triangles, SUW and SYW, that share fold SW. Which reason fills the blank?
- HL
- SAS
- AAS
- SSA
Jun opens garden shears so the blades HU and YD cross at their middles, F. Which statement fills the blank?
- △HFD ≅ △YFU
- △HFD ≅ △HFY
- △HFD ≅ △UFY
- HD ≅ UY
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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.
Kira draws a bike rack where bars HU and YD cross at F, and HD ∥ YU. Which statement fills the blank?
- ∠DHF ≅ ∠YUF
- ∠DHF ≅ ∠UYF
- DF ≅ YF
- ∠DHF ≅ ∠HDF
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.
Milo cuts a sandwich into two right triangles, GCS and GWS, along the shared cut GS. Which statement fills the blank?
- GC ≅ GS
- GC ≅ CS
- CS ≅ WS
- GS ≅ GS
Noah crosses craft sticks WD and GC at S, with WC ∥ GD and WC ≅ DG. Which statement fills the blank?
- ∠WSC ≅ ∠WSG
- ∠WSC ≅ ∠DSG
- ∠WSC ≅ ∠SWC
- CS ≅ GS
Pia draws scissors whose blades YC and VF cross at H, with YF ∥ VC. Which statement fills the blank?
- YH ≅ CH
- FH ≅ VH
- YC ≅ VF
- YH ≅ FH