Congruence proofs: fill in the missing step or reason

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Sketch, not to scale: segments UH and SQ cross at G (U upper left, H lower right, Q upper right, S lower left). Marked: UG and GH with one tick each; QG and GS with two ticks each. Given: G is the midpoint of UH and of SQ. Prove: UQ ∥ SH. Ladder: 1. G is the midpoint of UH -- Given; 2. UG ≅ HG -- Definition of midpoint; 3. G is the midpoint of SQ -- Given; 4. QG ≅ SG -- Definition of midpoint; 5. ∠UGQ ≅ ∠HGS -- Vertical angles theorem; 6. triangle UGQ ≅ triangle HGS -- SAS; 7. ∠QUG ≅ ∠SHG -- CPCTC; 8. UQ ∥ SH -- ?.

2. 1.

   Ren shows that bike-rack bars UQ and SH are parallel, using bars UH and SQ that cross at G. Which reason fills the blank?

   • Given
   • Alternate interior angles theorem
   • Converse of the alternate interior angles theorem
   • Converse of the corresponding angles postulate

3. Use this

   Given: V is the midpoint of CW and of QY. Prove: CY ∥ QW. Ladder: 1. V is the midpoint of CW -- Given; 2. CV ≅ WV -- Definition of midpoint; 3. V is the midpoint of QY -- Given; 4. YV ≅ QV -- Definition of midpoint; 5. ∠CVY ≅ ∠WVQ -- Vertical angles theorem; 6. triangle CVY ≅ triangle WVQ -- SAS; 7. ∠YCV ≅ ∠QWV -- CPCTC; 8. CY ∥ QW -- ?.

4. 2.

   Tala shows that frame edges CY and QW are parallel, using sticks CW and QY that cross at V. Which reason fills the blank?

   • Converse of the alternate interior angles theorem
   • Alternate interior angles theorem
   • Converse of the corresponding angles postulate
   • Given

5. Use this

   Given: D is the midpoint of FS and of HG. Prove: FG ∥ HS. Ladder: 1. D is the midpoint of FS -- Given; 2. FD ≅ SD -- Definition of midpoint; 3. D is the midpoint of HG -- Given; 4. GD ≅ HD -- Definition of midpoint; 5. ∠FDG ≅ ∠SDH -- Vertical angles theorem; 6. triangle FDG ≅ triangle SDH -- SAS; 7. ∠GFD ≅ ∠HSD -- ?; 8. FG ∥ HS -- Converse of the alternate interior angles theorem.

6. 3.

   Zee opens scissors with blades FS and HG crossing at D, and shows that FG ∥ HS. Which reason fills the blank?

   • CPCTC
   • Alternate interior angles theorem
   • Vertical angles theorem
   • SAS

7. Use this

   Given: W is the midpoint of VQ and of DU. Prove: VU ∥ DQ. Ladder: 1. W is the midpoint of VQ -- Given; 2. VW ≅ QW -- Definition of midpoint; 3. W is the midpoint of DU -- Given; 4. UW ≅ DW -- Definition of midpoint; 5. ∠VWU ≅ ∠QWD -- Vertical angles theorem; 6. triangle VWU ≅ triangle QWD -- SAS; 7. ∠UVW ≅ ∠DQW -- ?; 8. VU ∥ DQ -- Converse of the alternate interior angles theorem.

8. 4.

   Ari ties two tent cords VQ and DU that cross at W, and shows that VU ∥ DQ. Which reason fills the blank?

   • SAS
   • Alternate interior angles theorem
   • CPCTC
   • Vertical angles theorem
9. 5.

   Bo shows that a tent's two sides are equal when its pole stands at the middle of the floor. Which statement fills the blank?

   • CF ≅ DG
   • CD ≅ CG
   • ∠CFD ≅ ∠FCD
   • ∠CFD ≅ ∠CFG
10. 6.

    Esme shows that a roof's two slopes are equal when its post stands at the middle of the beam. Which statement fills the blank?

    • SU ≅ SW
    • SY ≅ UW
    • ∠SYU ≅ ∠SYW
    • ∠SYU ≅ ∠YSU
11. 7.

    Jun shows that a sign's two slanted edges are equal, using the line from the top to the middle of the base. Which statement fills the blank?

    • △HDQ ≅ △HDV
    • HQ ≅ HV
    • △HDQ ≅ △VDH
    • △HQV ≅ △HDV
12. 8.

    Luz proves that a tent's two sides YF and YW are equal, using pole YQ. Which statement and reason fill the blanks?

    • △YQF ≅ △YQW — HL
    • YF ≅ YW — CPCTC
    • △YQF ≅ △YQW — ASA
    • △YQF ≅ △YQW — SAS
13. 9.

    Milo proves that a roof's two slopes VH and VC are equal, using post VS. Which statement and reason fill the blanks?

    • △VSH ≅ △VSC — SAS
    • △VSH ≅ △VSC — HL
    • △VSH ≅ △VSC — ASA
    • VH ≅ VC — CPCTC
14. 10.

    Pia proves that a roof's two slopes QU and QG are equal, using post QH. Which statement and reason fill the blanks?

    • ∠QHU ≅ ∠QHG — Right angles congruence theorem
    • ∠QHU ≅ ∠QHG — CPCTC
    • ∠QHU ≅ ∠HQU — Right angles congruence theorem
    • ∠QHU ≅ ∠QHG — Vertical angles theorem