Use congruent triangles to find missing sides and angles

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

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

   • 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
   • AM = BM, CM = DM, ∠AMC = ∠BMD (vertical); ASA gives △AMC ≅ △BMD; AC = BD by CPCTC
2. 2.

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

3. 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.

4. 3.

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

   • ∠ADB = ∠ADC = 90°, since AD looks perpendicular
   • BD = CD (definition of midpoint)
   • ∠ABD = ∠ACD, pairing the wrong vertices
   • AD = AD (reflexive property)
5. 4.

   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 = √17 and nothing else
   • AB ∥ DC and AB = DC: both slopes are 1/4 and both lengths are √17
   • AB ∥ DC: both slopes are 1/4 and nothing else
6. 5.

   △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?

   • Their angles match, so AAA makes them congruent
   • Both are right triangles, so they are congruent
   • They look the same size on the grid
   • A 90° turn about the origin maps A, B, C to A′, B′, C′; turns keep lengths and angles
7. 6.

   △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?

   • Similar when a rigid motion maps one onto the other
   • 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
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 = 7, BC = 5, ∠A = 30° (opposite BC): AC is about 9.6 or 2.5
   • Right triangles with hypotenuse 13 and one leg 5: two triangles fit
   • 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
9. 8.

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

   • Right triangles with legs 6 and 4, and legs 8 and 3: both have area 12
   • 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 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 3, 4, 5 and sides 5, 12, 13: both perimeters are 12
    • Sides 5, 5, 6 and sides 4, 6, 6: both perimeters are 16
    • Sides 3, 4, 5 and sides 6, 8, 10: they look alike
    • Sides 5, 5, 6 and sides 6, 5, 5: both perimeters are 16
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 5, 5, 6 and sides 5, 5, 8: two sides match in each
    • Sides 3, 4, 5 and sides 4, 5, 6: both look nearly right-angled