Quantitative Aptitude

Geometry: Lines and Triangles: Practice Questions with Answers

24 practice questions on Geometry: Lines and Triangles, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.

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1. Q.1

   Two parallel lines are cut by a transversal. One of the angles formed is 70°. Find (a) its corresponding angle on the other parallel line, and (b) the co-interior angle adjacent to that corresponding angle.

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   Answer: (a) 70° (b) 110°

   Solution:

   1. (a) Corresponding angles are equal, so the corresponding angle is also 70°.
   2. (b) Co-interior angles add up to 180°, so the adjacent co-interior angle = 180° − 70° = 110°.

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2. Q.2

   In triangle ABC, angle A = 50° and angle B = 70°. Find angle C.

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   Answer: 60°

   Solution:

   1. The three angles of a triangle sum to 180°: angle C = 180° − 50° − 70° = 60°.

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3. Q.3

   In an isosceles triangle, the vertex angle is 40°. Find each of the two base angles.

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   Answer: 70° each

   Solution:

   1. The two base angles are equal. Their combined measure = 180° − 40° = 140°.
   2. Each base angle = 140° / 2 = 70°.

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4. Q.4

   Can a triangle have sides of length 4 cm, 5 cm and 10 cm?

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   Answer: No — 4 + 5 = 9 < 10, so the triangle inequality fails

   Solution:

   1. Check the triangle inequality: the sum of the two smaller sides must exceed the largest side.
   2. 4 + 5 = 9, which is less than 10.
   3. Since the inequality fails, no such triangle can exist.

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5. Q.5

   The exterior angle of a triangle is 110°, and one of the interior opposite angles is 40°. Find the other interior opposite angle.

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   Answer: 70°

   Solution:

   1. The exterior angle equals the sum of the two non-adjacent interior angles: 110° = 40° + (other angle), so the other angle = 70°.

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6. Q.6

   Two angles of a triangle are in the ratio 2:3, and the third angle is 100°. Find the two angles.

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   Answer: 32° and 48°

   Solution:

   1. Let the two angles be 2x and 3x: 2x + 3x + 100° = 180°, so 5x = 80°, giving x = 16°.
   2. The two angles are 2(16°) = 32° and 3(16°) = 48°.

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7. Q.7

   In a triangle, the three angles are in the ratio 1:2:3. Find the largest angle.

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   Answer: 90°

   Solution:

   1. Let the angles be x, 2x and 3x: x + 2x + 3x = 180°, so 6x = 180°, giving x = 30°.
   2. The largest angle is 3(30°) = 90°.

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8. Q.8

   In triangle PQR, angle P = 65° and angle Q = 55°. Find angle R.

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   Answer: 60°

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9. Q.9

   In an isosceles triangle, the vertex angle is 50°. Find each base angle.

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   Answer: 65° each. (180° − 50°) / 2.

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10. Q.10

    The angles of a triangle are in the ratio 2:3:5. Find all three angles.

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    Answer: 36°, 54° and 90°. 10x = 180° gives x = 18°.

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11. Q.11

    Two parallel lines are cut by a transversal. One angle formed is 110°. Find its alternate angle and its co-interior angle.

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    Answer: Alternate = 110° (equal); co-interior = 70° (180° − 110°).

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12. Q.12

    Two angles of a triangle are in the ratio 3:4, and the third angle is 40°. Find the two angles.

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    Answer: 60° and 80°

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13. Q.13

    Can a triangle have sides of length 6 cm, 7 cm and 14 cm?

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    Answer: No. 6 + 7 = 13 < 14, so the triangle inequality fails.

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14. Q.14

    The exterior angle of a triangle is 125°, and one of the interior opposite angles is 55°. Find the other interior opposite angle.

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    Answer: 70°

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15. Q.15

    A right triangle has hypotenuse 30 cm. Find the radius of its circumcircle in cm.

    1. A 15
    2. B 30
    3. C 7.5
    4. D 16
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    Correct answer: (A) 15

    The circumcentre of a right triangle is the hypotenuse midpoint. The circumradius is therefore half the hypotenuse, 30/2 = 15 cm. Check: the hypotenuse is a diameter, so twice this radius is 30 cm.

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16. Q.16

    I is the incentre of triangle ABC and angle A = 50°. Find the smaller angle BIC in degrees.

    1. A 115
    2. B 130
    3. C 65
    4. D 50
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    Correct answer: (A) 115

    In triangle BIC, the other two angles are B/2 and C/2. Their sum is (180 − 50)/2, so BIC = 180 − (180 − 50)/2 = 115°. Check with the equivalent form 90° + A/2 = 90° + 25°.

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17. Q.17

    A right triangle has hypotenuse 46 cm. Find the radius of its circumcircle in cm.

    1. A 23
    2. B 46
    3. C 11.5
    4. D 24
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    Correct answer: (A) 23

    The circumcentre of a right triangle is the hypotenuse midpoint. The circumradius is therefore half the hypotenuse, 46/2 = 23 cm. Check: the hypotenuse is a diameter, so twice this radius is 46 cm.

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18. Q.18

    G is the centroid of triangle ABC and AM is a median of length 24 cm. Find AG in cm.

    1. A 16
    2. B 8
    3. C 12
    4. D 24
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    Correct answer: (A) 16

    The centroid divides a median in ratio AG:GM = 2:1. Thus AG = 2/3 × 24 = 16 cm. Check: GM = 8; AG is twice GM and their sum is 24.

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19. Q.19

    I is the incentre of triangle ABC and angle A = 42°. Find the smaller angle BIC in degrees.

    1. A 138
    2. B 69
    3. C 111
    4. D 42
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    Correct answer: (C) 111

    In triangle BIC, the other two angles are B/2 and C/2. Their sum is (180 − 42)/2, so BIC = 180 − (180 − 42)/2 = 111°. Check with the equivalent form 90° + A/2 = 90° + 21°.

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20. Q.20

    G is the centroid of triangle ABC and AM is a median of length 36 cm. Find AG in cm.

    1. A 12
    2. B 24
    3. C 18
    4. D 36
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    Correct answer: (B) 24

    The centroid divides a median in ratio AG:GM = 2:1. Thus AG = 2/3 × 36 = 24 cm. Check: GM = 12; AG is twice GM and their sum is 36.

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21. Q.21

    A right triangle has hypotenuse 38 cm. Find the radius of its circumcircle in cm.

    1. A 38
    2. B 9.5
    3. C 19
    4. D 20
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    Correct answer: (C) 19

    The circumcentre of a right triangle is the hypotenuse midpoint. The circumradius is therefore half the hypotenuse, 38/2 = 19 cm. Check: the hypotenuse is a diameter, so twice this radius is 38 cm.

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22. Q.22

    I is the incentre of triangle ABC and angle A = 58°. Find the smaller angle BIC in degrees.

    1. A 122
    2. B 119
    3. C 61
    4. D 58
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    Correct answer: (B) 119

    In triangle BIC, the other two angles are B/2 and C/2. Their sum is (180 − 58)/2, so BIC = 180 − (180 − 58)/2 = 119°. Check with the equivalent form 90° + A/2 = 90° + 29°.

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23. Q.23

    Triangle ABC is right-angled at A, with AB = 24 cm and AC = 32 cm. Which vertex is its orthocentre?

    1. A B
    2. B C
    3. C A
    4. D No vertex
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    Correct answer: (C) A

    AB is perpendicular to AC. Hence the altitude from B lies on BA and the altitude from C lies on CA; they intersect at A. Check: the orthocentre is the intersection of the altitudes, so the right-angle vertex A is the required point.

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24. Q.24

    Triangle ABC is right-angled at A, with AB = 12 cm and AC = 16 cm. Which vertex is its orthocentre?

    1. A B
    2. B C
    3. C No vertex
    4. D A
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    Correct answer: (D) A

    AB is perpendicular to AC. Hence the altitude from B lies on BA and the altitude from C lies on CA; they intersect at A. Check: the orthocentre is the intersection of the altitudes, so the right-angle vertex A is the required point.

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