Triangles – Theorems Quiz - ICSE Class 9 Mathematics

Free practice quiz: Triangles – Theorems (ICSE Class 9 Mathematics). 53 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 9 Mathematics practice quiz has 53 multiple-choice questions on Triangles – Theorems, and 53 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. In triangle ABC, if AB = AC and angle BAC = 80 degrees, then what is the measure of angle ABC?

   1. 40 degrees
   2. 50 degrees
   3. 60 degrees
   4. 70 degrees
   Show answer

   Answer: B. 50 degrees

   Since AB = AC, triangle ABC is an isosceles triangle. The angles opposite to equal sides are equal, so angle ABC = angle ACB. The sum of angles in a triangle is 180 degrees. Therefore, angle ABC + angle ACB + angle BAC = 180 degrees. 2 * angle ABC + 80 degrees = 180 degrees. 2 * angle ABC = 100 degrees. Angle ABC = 50 degrees.

2. In a triangle ABC, if angle A > angle B, then which side is longer?

   1. BC
   2. AC
   3. AB
   4. None of the above
   Show answer

   Answer: A. BC

   The theorem states that in any triangle, the side opposite the larger angle is longer than the side opposite the smaller angle. Since angle A is greater than angle B, the side opposite angle A (which is BC) is longer than the side opposite angle B (which is AC).

3. If triangle ABC is congruent to triangle DEF (ABC ≅ DEF), which angle is equal to angle B?

   1. Angle D
   2. Angle E
   3. Angle F
   4. None of the above
   Show answer

   Answer: B. Angle E

   When two triangles are congruent, their corresponding angles are equal. In the congruence statement ABC ≅ DEF, the order of vertices is important. Angle A corresponds to Angle D, Angle B corresponds to Angle E, and Angle C corresponds to Angle F. Therefore, Angle B = Angle E.

4. If triangle ABC and triangle DBC share the same base BC and lie between the same parallels AB and DC, then what is the relationship between their areas?

   1. Area(ABC) < Area(DBC)
   2. Area(ABC) > Area(DBC)
   3. Area(ABC) = Area(DBC)
   4. Area(ABC) is twice Area(DBC)
   Show answer

   Answer: C. Area(ABC) = Area(DBC)

   Triangles on the same base and between the same parallels have equal areas. Since triangle ABC and triangle DBC share base BC and are between parallels AB and DC (assuming AB || DC), their heights with respect to the base BC are equal. Hence, their areas are equal.

5. Which of the following is NOT a criterion for congruence of triangles?

   1. SSS
   2. SAS
   3. AAA
   4. ASA
   Show answer

   Answer: C. AAA

   SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle) are valid congruence criteria. AAA (Angle-Angle-Angle) is not a congruence criterion because triangles with equal angles can be similar but not necessarily congruent (they can be of different sizes).

More questions in this quiz

Answers and explanations are in the full quiz.

1. If two angles of a triangle are 30 degrees and 70 degrees, what is the measure of the third angle?
2. In an equilateral triangle, what is the measure of each angle?
3. Which theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the longest side (hypotenuse)?
4. If a triangle has sides of length 5 cm, 12 cm, and 13 cm, what type of triangle is it?
5. In triangle PQR, if PQ = PR and angle PQR = 45 degrees, what is the measure of angle QPR?
6. The sum of any two sides of a triangle is always ______ than the third side.
7. If triangle XYZ is congruent to triangle ABC (XYZ ≅ ABC), which of the following is necessarily true?
8. In triangle DEF, if DE = EF = FD, then which property holds true for its angles?
9. If two triangles have corresponding angles equal, they are said to be:
10. If the hypotenuse of a right-angled triangle is 15 cm and one of the other sides is 9 cm, what is the length of the third side?
11. Which theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, the other two sides are divided in the same ratio?
12. In triangle ABC, if AD is the angle bisector of angle A and D lies on BC. According to the Angle Bisector Theorem, which of the following is true?
13. If two triangles are congruent, then their areas are:
14. In triangle MNP, if MP = NP and angle MPN = 120 degrees, what is the measure of angle PMN?
15. If the angles of a triangle are in the ratio 1:2:3, what are the measures of the angles?
16. Which theorem states that a line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side?
17. In triangle ABC, if angle A = 90 degrees and AB = 6 cm, AC = 8 cm, what is the length of BC?
18. If two triangles are similar, then the ratio of their corresponding sides is equal to the ratio of their:
19. In an isosceles triangle, if the vertex angle is 100 degrees, what is the measure of each of the base angles?
20. In a right-angled triangle, if one acute angle is 35 degrees, what is the measure of the other acute angle?
21. If triangle ABC has sides AB=5, BC=7, AC=9, which statement about the angles is correct?
22. If two triangles have all three pairs of corresponding sides equal (SSS congruence criterion), then the triangles are:
23. In triangle ABC, if AD is the median to side BC, which statement is true regarding the lengths of BD and CD?
24. If a triangle is right-angled, then the square of the hypotenuse is equal to the sum of the squares of the other two sides. This statement is known as:
25. In triangle ABC, if AB || DE and D is the midpoint of AC and E is the midpoint of BC, what is the relationship between DE and AB?
26. In triangle ABC, if AB = AC and angle ABC = 50 degrees, what is the measure of angle BAC?
27. If triangle PQR is similar to triangle XYZ (PQR ~ XYZ), and PQ = 4 cm, QR = 6 cm, RP = 8 cm, and XY = 8 cm, what is the length of YZ?
28. In a triangle, if one side is extended, the exterior angle formed is equal to the sum of the two interior opposite angles. This is known as the:
29. In triangle ABC, if angle A = 60 degrees and angle B = 70 degrees, what is the measure of angle C?
30. If triangle ABC is congruent to triangle PQR by ASA criterion, and angle A = 50 degrees, AB = 6 cm, angle B = 70 degrees, then what is the length of PQ?
31. In triangle ABC, if D is a point on BC such that AD bisects angle BAC and AB/AC = BD/CD. What is the nature of AD?
32. Which of the following sets of side lengths CANNOT form a triangle?
33. In triangle ABC, if AB = BC, then which angles are equal?
34. If triangle ABC ~ triangle PQR, and angle A = 40 degrees, angle Q = 60 degrees, what is the measure of angle R?
35. In triangle ABC, if AD is the altitude to BC, then which of the following is true for a right-angled triangle at A?
36. If two triangles have corresponding sides in proportion (SSS similarity criterion), then the triangles are:
37. In triangle ABC, if AB=AC and angle ABC = 50 degrees, then what is angle BAC?
38. In triangle ABC, if AD is the angle bisector of angle A, and AB/AC = BD/CD, and AB=6, AC=8, BD=3, then what is the length of CD?
39. If two parallel lines are intersected by a transversal, then the alternate interior angles are:
40. In triangle ABC, if AD is the median to BC, then which statement is true?
41. In triangle ABC, if angle A = 80 degrees and angle B = 40 degrees, what is the measure of angle C?
42. If triangle ABC is congruent to triangle XYZ by SSS criterion, and AB=5, BC=7, AC=9, XY=5, YZ=7, ZX=9, then which of the following is true?
43. In triangle ABC, if AB = AC and angle ABC = 70 degrees, what is the measure of angle BAC?
44. If a triangle has sides of length 3 cm, 5 cm, and 7 cm, what type of triangle is it?
45. Which theorem states that if a line is drawn through the midpoint of one side of a triangle parallel to another side, it bisects the third side?
46. In triangle ABC, if angle B = 90 degrees and AB = 5 cm, BC = 12 cm, what is the length of AC?
47. If two triangles have two pairs of corresponding sides proportional and the included angles equal, then the triangles are:
48. In triangle ABC, if AD is the median to BC, then AB^2 + AC^2 is related to AD and BD by which Apollonius' Theorem relationship?

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