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?

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

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

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

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

   • △ABC ≅ △ACD by ASA, so AB = AC by CPCTC
   • 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
   • Opposite sides of a parallelogram are equal, so AB = DC at once

4. 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.

5. 4.

   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
6. 5.

   For A(1, 1), B(5, 2), C(6, 6) and D(2, 5), Luz proves ABCD is a parallelogram. Which computed fact is enough on its own?

   • Not a parallelogram, because the diagonals are unequal
   • AB ∥ DC: both slopes are 1/4 and nothing else
   • AB = DC = √17 and nothing else
   • AB ∥ DC and AB = DC: both slopes are 1/4 and both lengths are √17
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?

   • 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
   • They look the same size on the grid
8. 7.

   △ABC has A(0, 0), B(4, 0), C(0, 3). △A′B′C′ has A′(0, 0), B′(8, 0), C′(0, 6). Noah compares them. Which statement is justified?

   • A dilation from the origin with scale factor 2 maps one to the other: similar, not congruent
   • Not similar: the side lengths differ
   • Congruent: both are right triangles at the origin
   • Similar when a rigid motion maps one onto the other
9. 8.

   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
   • AB = 7, AC = 5 and ∠A = 30° (between the sides): the triangle is not fixed
   • 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
10. 9.

    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 8 and 3: both have area 12
    • Right triangles with legs 6 and 4, and legs 4 and 6: both have area 12
    • Right triangles with hypotenuse 10 and area 24 in both
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 4, 5, 6: both look nearly right-angled
    • Sides 3, 4, 5 and sides 4, 3, 5: same angles, same size
    • Sides 5, 5, 6 and sides 5, 5, 8: two sides match in each
    • Sides 3, 4, 5 and sides 6, 8, 10: same angles, different sizes