Question

reasoning in geometry (2) topic test
instructions · this part consists of 12 multiple - choice questions.
· fill in only one circle for each question.
· each question is worth 1 mark.
· calculators are not allowed.
time allowed: 15 minutes
total marks: 12
1 the angle sum of a triangle is always equal to
(a) 90° (b) 180° (c) 270° (d) 360°
2 the angle sum of a quadrilateral is always equal to
(a) 90° (b) 180° (c) 270° (d) 360°
3 if two angles of a triangle are complementary then the third angle is
(a) acute (b) right angle (c) obtuse (d) none of these
4 if two angles of a triangle are 30° and 40° then the third angle is equal to
(a) 20° (b) 110° (c) 200° (d) 290°
5 in any right - angled triangle the other two angles are both
(a) acute (b) obtuse (c) right angles (d) none of these
6 the size of each angle of an equilateral triangle is
(a) 30° (b) 60° (c) 90° (d) 45°
7 in a right - angled triangle if one acute angle is 35°, the size of the other acute angle is
(a) 25° (b) 35° (c) 45° (d) 55°
8 the size of each of the base angles of a right - angled isosceles triangle is
(a) 45° (b) 60° (c) 90° (d) 120°
9 if the angles of a rhombus are x and 2x, then x is
(a) acute (b) right angle (c) obtuse (d) reflex
10 if the angle between the two equal sides of an isosceles triangle is 120°, the size of each remaining angle is
(a) 20° (b) 30° (c) 60° (d) 80°
11 in a triangle, the number of right angles can be
(a) 1 (b) 2 (c) 3 (d) none of these
12 in a triangle, if one angle is obtuse then the other two angles are
(a) acute (b) obtuse (c) right angle (d) reflex

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is always \(180^{\circ}\). So for question 1, the answer is B.

Step2: Recall angle - sum property of a quadrilateral

The sum of interior angles of a quadrilateral is \(360^{\circ}\). So for question 2, the answer is D.

Step3: Use complementary - angle concept in a triangle

If two angles of a triangle are complementary (sum to \(90^{\circ}\)), then the third angle is \(180 - 90=90^{\circ}\) (a right - angle). So for question 3, the answer is B.

Step4: Apply angle - sum property of a triangle

If two angles of a triangle are \(30^{\circ}\) and \(40^{\circ}\), then the third angle is \(180-(30 + 40)=110^{\circ}\). So for question 4, the answer is B.

Step5: Analyze angles in a right - angled triangle

In a right - angled triangle, one angle is \(90^{\circ}\), and the sum of the other two angles is \(180 - 90 = 90^{\circ}\), so they are both acute. So for question 5, the answer is A.

Step6: Calculate angles in an equilateral triangle

In an equilateral triangle, all angles are equal. Let each angle be \(x\). Then \(x+x + x=180^{\circ}\), so \(x = 60^{\circ}\). So for question 6, the answer is B.

Step7: Find the other acute angle in a right - angled triangle

In a right - angled triangle, if one acute angle is \(35^{\circ}\), and the sum of the two acute angles is \(90^{\circ}\), then the other acute angle is \(90 - 35=55^{\circ}\). So for question 7, the answer is D.

Step8: Determine base angles of a right - angled isosceles triangle

In a right - angled isosceles triangle, one angle is \(90^{\circ}\), and the other two equal angles sum to \(180 - 90=90^{\circ}\), so each base angle is \(45^{\circ}\). So for question 8, the answer is A.

Step9: Analyze angles of a rhombus

In a rhombus, adjacent angles are supplementary (\(x + 2x=180^{\circ}\), so \(x = 60^{\circ}\), which is acute). So for question 9, the answer is A.

Step10: Calculate angles of an isosceles triangle

In an isosceles triangle, if the angle between the two equal sides is \(120^{\circ}\), and the sum of interior angles is \(180^{\circ}\), then the sum of the other two equal angles is \(180 - 120 = 60^{\circ}\), so each of the remaining angles is \(30^{\circ}\). So for question 10, the answer is B.

Step11: Consider the number of right angles in a triangle

The sum of interior angles of a triangle is \(180^{\circ}\), so a triangle can have at most 1 right angle. So for question 11, the answer is A.

Step12: Analyze angles in a triangle with an obtuse angle

If one angle of a triangle is obtuse (greater than \(90^{\circ}\)), then the sum of the other two angles must be less than \(90^{\circ}\) (since the sum of all three angles is \(180^{\circ}\)), so they are acute. So for question 12, the answer is A.

Answer:

1. B. \(180^{\circ}\)
2. D. \(360^{\circ}\)
3. B. right angle
4. B. \(110^{\circ}\)
5. A. acute
6. B. \(60^{\circ}\)
7. D. \(55^{\circ}\)
8. A. \(45^{\circ}\)
9. A. acute
10. B. \(30^{\circ}\)
11. A. 1
12. A. acute