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?

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

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

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

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?

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

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

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

   • Not similar: the side lengths differ
   • Congruent: both are right triangles at the origin
   • 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
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
   • 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 = 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
   • Right triangles with hypotenuse 10 and area 24 in both
   • 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
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 6, 5, 5: both perimeters are 16
    • Sides 3, 4, 5 and sides 6, 8, 10: they look alike
    • Sides 5, 5, 6 and sides 4, 6, 6: both perimeters are 16
    • Sides 3, 4, 5 and sides 5, 12, 13: both perimeters are 12
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, 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
    • Sides 3, 4, 5 and sides 6, 8, 10: same angles, different sizes