Triangle proofs: fill in the missing step or reason

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Use this

   Given: triangle CDF. Prove: m∠CDF + m∠DCF + m∠CFD = 180. Ladder: 1. GH ∥ DF, through C -- Parallel postulate; 2. ∠GCD ≅ ∠CDF and ∠FCH ≅ ∠CFD -- Alternate interior angles theorem; 3. m∠GCD = m∠CDF and m∠FCH = m∠CFD -- Definition of congruent angles; 4. m∠GCD + m∠DCF + m∠FCH = 180 -- Angle addition postulate and definition of a straight angle; 5. m∠CDF + m∠DCF + m∠CFD = 180 -- ?.

2. 1.

   Jun proves the triangle angle sum theorem on a sail CDF. Which reason fills the blank?

   • Angle addition postulate
   • Triangle angle sum theorem
   • Transitive property
   • Substitution property

3. Use this

   Given: triangle QUW. Prove: m∠QUW + m∠UQW + m∠QWU = 180. Ladder: 1. SY ∥ UW, through Q -- Parallel postulate; 2. ∠SQU ≅ ∠QUW and ∠WQY ≅ ∠QWU -- Alternate interior angles theorem; 3. m∠SQU = m∠QUW and m∠WQY = m∠QWU -- Definition of congruent angles; 4. m∠SQU + m∠UQW + m∠WQY = 180 -- Angle addition postulate and definition of a straight angle; 5. m∠QUW + m∠UQW + m∠QWU = 180 -- ?.

4. 2.

   Kira proves the triangle angle sum theorem on a pizza slice QUW. Which reason fills the blank?

   • Triangle angle sum theorem
   • Definition of supplementary angles
   • Symmetric property
   • Substitution property

5. Use this

   Given: triangle SUV, with UV extended to W. Prove: m∠SUV + m∠USV = m∠SVW. Ladder: 1. triangle SUV, with UV extended to W -- Given; 2. m∠SUV + m∠USV + m∠SVU = 180 -- Triangle angle sum theorem; 3. ∠SVU and ∠SVW are supplementary -- Linear pair postulate; 4. m∠SVU + m∠SVW = 180 -- Definition of supplementary angles; 5. m∠SUV + m∠USV + m∠SVU = m∠SVU + m∠SVW -- Substitution property; 6. m∠SUV + m∠USV = m∠SVW -- ?.

6. 3.

   Pia proves the exterior angle theorem on a road sign SUV, extending the bottom edge UV to W. Which reason fills the blank?

   • Addition property of equality
   • Subtraction property of equality
   • Given
   • Exterior angle theorem

7. Use this

   Given: DF ≅ DG, and Q is the midpoint of FG. Prove: ∠DFQ ≅ ∠DGQ. Ladder: 1. DF ≅ DG -- Given; 2. Q is the midpoint of FG -- Given; 3. FQ ≅ GQ -- Definition of midpoint; 4. DQ ≅ DQ -- Reflexive property; 5. triangle DFQ ≅ triangle DGQ -- SSS; 6. ∠DFQ ≅ ∠DGQ -- ?.

8. 4.

   Zee proves the base angles theorem on a tent front DFG, with a pole from D to the middle Q of the floor. Which reason fills the blank?

   • Base angles theorem
   • SSS
   • CPCTC
   • Given

9. Use this

   Given: SU ≅ SW, and Y is the midpoint of UW. Prove: ∠SUY ≅ ∠SWY. Ladder: 1. SU ≅ SW -- Given; 2. Y is the midpoint of UW -- Given; 3. UY ≅ WY -- Definition of midpoint; 4. SY ≅ SY -- Reflexive property; 5. triangle SUY ≅ triangle SWY -- SSS; 6. ∠SUY ≅ ∠SWY -- ?.

10. 5.

    Ari proves the base angles theorem on a roof SUW, with a beam from S to the middle Y of UW. Which reason fills the blank?

    • Base angles theorem
    • Given
    • SSS
    • CPCTC

11. Sketch, not to scale

    Triangle GSY with G at the top, S bottom left and Y bottom right; line UW is drawn through G (U on the left, W on the right). Marked: parallel arrows on UW and SY. Given: △GSY. Prove: m∠GSY + m∠SGY + m∠GYS = 180. Ladder: 1. UW ∥ SY, through G — Parallel postulate; 2. ? — Alternate interior angles theorem; 3. m∠UGS = m∠GSY and m∠YGW = m∠GYS — Definition of congruent angles; 4. m∠UGS + m∠SGY + m∠YGW = 180 — Angle addition postulate and definition of a straight angle; 5. m∠GSY + m∠SGY + m∠GYS = 180 — Substitution property.

12. 6.

    Milo proves the triangle angle sum theorem on a tent front GSY. Which statement fills the blank?

    • m∠GSY + m∠SGY + m∠GYS = 180
    • ∠UGS ≅ ∠GYS and ∠YGW ≅ ∠GSY
    • ∠UGS ≅ ∠GSY and ∠YGW ≅ ∠GYS
    • ∠UGS ≅ ∠SGY and ∠YGW ≅ ∠SGY

13. Use this

    Given: triangle HQY, with QY extended to D. Prove: m∠HQY + m∠QHY = m∠HYD. Ladder: 1. triangle HQY, with QY extended to D -- Given; 2. m∠HQY + m∠QHY + m∠HYQ = 180 -- Triangle angle sum theorem; 3. ? -- Linear pair postulate; 4. m∠HYQ + m∠HYD = 180 -- Definition of supplementary angles; 5. m∠HQY + m∠QHY + m∠HYQ = m∠HYQ + m∠HYD -- Substitution property; 6. m∠HQY + m∠QHY = m∠HYD -- Subtraction property of equality.

14. 7.

    Ren proves the exterior angle theorem on a ramp side HQY, with the floor running on to D. Which statement fills the blank?

    • ∠HYQ and ∠HYD are supplementary
    • ∠HYQ ≅ ∠HYD
    • m∠HQY + m∠QHY = m∠HYD
    • ∠HYQ and ∠HYD are complementary

15. Use this

    Sketch, not to scale: triangle GCW with G at the top, C bottom left and W bottom right; side CW is extended past W to F. Marked: the outside angle ∠GWF shaded. Given: triangle GCW, with CW extended to F. Prove: m∠GCW + m∠CGW = m∠GWF. Ladder: 1. triangle GCW, with CW extended to F -- Given; 2. m∠GCW + m∠CGW + m∠GWC = 180 -- Triangle angle sum theorem; 3. ? -- Linear pair postulate; 4. m∠GWC + m∠GWF = 180 -- Definition of supplementary angles; 5. m∠GCW + m∠CGW + m∠GWC = m∠GWC + m∠GWF -- Substitution property; 6. m∠GCW + m∠CGW = m∠GWF -- Subtraction property of equality.

16. 8.

    Tala proves the exterior angle theorem on a sandwich half GCW, with edge CW drawn on to F. Which statement fills the blank?

    • ∠GWC ≅ ∠GWF
    • m∠GCW + m∠CGW = m∠GWF
    • ∠GWC and ∠GWF are complementary
    • ∠GWC and ∠GWF are supplementary
17. 9.

    Bo proves the base angles theorem on a road sign CHV, marking the middle Q of HV. Which statement fills the blank?

    • △CHV ≅ △CVQ
    • △CHQ ≅ △CVQ
    • △CHQ ≅ △QVC
    • ∠CHQ ≅ ∠CVQ
18. 10.

    Esme proves the base angles theorem on a pyramid face FDY, marking the middle G of DY. Which statement fills the blank?

    • △FDG ≅ △FYG
    • △FDG ≅ △GYF
    • ∠FDG ≅ ∠FYG
    • △FDY ≅ △FYG