Use congruent triangles to find missing sides and angles

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   Ari proves CB = CD for the card. Which lines complete the proof?

   • CB and CD look equal; AC = AC; SSS gives △ABC ≅ △ADC
   • ∠BAC = ∠DAC, so CB = CD by CPCTC
   • AC = AC (reflexive); SAS gives △ABC ≅ △ADC; CB = CD by CPCTC
   • AC = AC (reflexive); ASA gives △ABC ≅ △ADC; CB = CD by CPCTC
2. 2.

   Bo proves AC = BD. Which lines complete the proof?

   • AM = BM, CM = DM, ∠AMC = ∠BMD (vertical); SAS gives △AMC ≅ △BMD; AC = BD by CPCTC
   • AC = BD is assumed, so △AMC ≅ △BMD by SSS
   • ∠AMC = ∠BMD (vertical), so AC = BD by CPCTC
   • AM = BM, CM = DM, ∠AMC = ∠BMD (vertical); ASA gives △AMC ≅ △BMD; AC = BD by CPCTC
3. 3.

   Esme proves AB = AC. Which lines complete the proof?

   • Two angle pairs match, so AB = AC by CPCTC
   • AD ⊥ BC in the picture, so △ABD ≅ △ACD by HL
   • AD = AD (reflexive); ASA gives △ABD ≅ △ACD; AB = AC by CPCTC
   • AD = AD (reflexive); AAS gives △ABD ≅ △ACD; AB = AC by CPCTC
4. 4.

   Jun proves AB = DC. Which lines complete the proof?

   • Opposite sides of a parallelogram are equal, so AB = DC at once
   • Alternate angles and AC = AC give SAS, so △ABC ≅ △CDA and AB = DC
   • Alternate angles and AC = AC give ASA, so △ABC ≅ △CDA and AB = DC
   • △ABC ≅ △ACD by ASA, so AB = AC by CPCTC

5. Read this first

   Use this

   Proof: 1. AB = AC (given). 2. ∠BAD = ∠CAD (AD bisects ∠BAC). 3. ? 4. triangle ABD ≅ triangle ACD (SAS). 5. BD = CD (CPCTC), so D is the midpoint.

6. 5.

   Kira proves D is the midpoint of BC. Which line is missing from the proof?

   • ∠ADB = ∠ADC = 90°, since AD looks perpendicular
   • AD = AD (reflexive property)
   • BD = CD (definition of midpoint)
   • ∠ABD = ∠ACD, pairing the wrong vertices
7. 6.

   △ABC has A(1, 1), B(4, 1), C(1, 3). △A′B′C′ has A′(−1, 1), B′(−1, 4), C′(−3, 1). Milo says they are congruent. Which reasoning proves it?

   • They look the same size on the grid
   • Both are right triangles, so they are congruent
   • Their angles match, so AAA makes them congruent
   • A 90° turn about the origin maps A, B, C to A′, B′, C′; turns keep lengths and angles
8. 7.

   Two triangles with two equal sides and one equal angle are always congruent. Pia looks for numbers that break the claim. Which pair of triangles does it?

   • AB = 5, BC = 7 and ∠A = 30° (opposite the longer side): two triangles fit
   • Right triangles with hypotenuse 13 and one leg 5: two triangles fit
   • AB = 7, BC = 5, ∠A = 30° (opposite BC): AC is about 9.6 or 2.5
   • AB = 7, AC = 5 and ∠A = 30° (between the sides): the triangle is not fixed
9. 8.

   Two triangles with equal areas are congruent. Ren looks for a pair that breaks the claim. Which pair does it?

   • Equilateral triangles with sides 4 and 4.1: the areas are equal
   • Right triangles with legs 6 and 4, and legs 4 and 6: both have area 12
   • Right triangles with legs 6 and 4, and legs 8 and 3: both have area 12
   • Right triangles with hypotenuse 10 and area 24 in both
10. 9.

    Two triangles with equal perimeters are congruent. Tala looks for a pair that breaks the claim. Which pair does it?

    • Sides 5, 5, 6 and sides 4, 6, 6: both perimeters are 16
    • Sides 5, 5, 6 and sides 6, 5, 5: both perimeters are 16
    • Sides 3, 4, 5 and sides 5, 12, 13: both perimeters are 12
    • Sides 3, 4, 5 and sides 6, 8, 10: they look alike
11. 10.

    Two triangles with all three angles equal are congruent. Zee looks for a pair that breaks the claim. Which pair does it?

    • Sides 3, 4, 5 and sides 6, 8, 10: same angles, different sizes
    • Sides 3, 4, 5 and sides 4, 3, 5: same angles, same size
    • Sides 3, 4, 5 and sides 4, 5, 6: both look nearly right-angled
    • Sides 5, 5, 6 and sides 5, 5, 8: two sides match in each