Olympiad Test: Geometrical Shapes And Angles - 2 - Class 5 MCQ

An acute angle is less than 90°. 91° is more than 90°, so it is not acute.

An acute angle is less than 90° i.e. it is less than a right angle. Half a right angle (45°) is less than a right angle, so it is acute.

The following angles are all less than 90°, so are acute:
∠AOB, ∠BOC,∠COD,  ∠DOE, ∠AOC, 
∠AOD, ∠BOD, ∠BOE and ∠COE
Therefore there are 9 acute angles altogether. Note that ∠AOE is obtuse.

Each of the five triangles at the points of the pentagram has three acute angles.
So there are 5 × 3 = 15 acute angles altogether in the pentagram. (All other angles are not acute)

The marked angles are all less than 90°, so are acute:

Therefore there are 10 acute angles altogether.

The angles marked with arcs are all less than 90°, so are acute. The four angles around the top point are all right angles.
So there are 10 acute angles altogether.

Because one pair of lines is parallel (that is what the two arrows mean) and the other two lines are perpendicular to them (the little boxes mean right angles), the shape inside the four lines must be a rectangle. And there are 4 right angles at each of the four points where lines intersect. So there are 4 × 4 = 16 right angles altogether.

A full rotation consists of four right angles. So two full rotations consist of eight right angles.

An obtuse angle is more than 90° but less than 180°.182° is more than 180°, so it is not obtuse.

The following angles are obtuse:
∠AOD, ∠AOE, ∠AOF, ∠BOD, ∠BOE, 
∠BOF, ∠COE and ∠COF.
Therefore, there are 8 obtuse angles altogether.

Half a right angle = 45°
One right angle = 90°
One and a half right angles = 1½ × 90° = 135° 
Two right angles = 180°
Only 135° is greater than 90° but less than 180°, so it is the only obtuse angle.

A full rotation consists of two straight (180°) angles.
So three full rotations consist of six straight angles.

A reflex angle is more than 180° but less than 360°. 178° is less than 180°,
so it is not reflex. (All the rest are reflex angles.)

A Reflex Angle is greater than 180° but less than 360°.
A, B and D are all less than 180°.
Only C is greater than 180° but less than 360°.

If you add the smaller angle (acute or obtuse) and the reflex angle for the same shape you will always come to 360°.
Therefore, reflex angle AOB = 360° − 67° = 293°

2 × 33° = 66°, which is between 0° and 90°, so acute.
5 × 33° = 165°, which is between 90° and 180°, so obtuse.
6 × 33° = 198°, which is between 180° and 360°, so reflex.
7 × 33° = 231°, which is also reflex, but we have already found the least amount which is 6.

The triangle has three different lengths of side, so is scalene.
The triangle has lengths 3, 4 and 5 and 52 = 32 + 42 , so is right angled.
Therefore, it is a scalene right angled triangle.

Since the triangle is isosceles, it must have two equal angles. Since it is also right angled, the other two angles must both be 45° (45° + 45° +90° = 180°) So A is true.
Since the triangle is isosceles, it must have two of its sides equal. So B is true.
An isosceles triangle has one line of symmetry. So C is also true.
A triangle that satisfies A, B and C is shown in the following diagram:

Line l is the axis of symmetry Statement D must be false since a triangle with three different sides cannot be isosceles.

Shraddha could make an isosceles triangle, a parallelogram, a rectangle or a kite:

But the rectangle is not a square because the sides have different lengths.

Shraddha could make a square, an isosceles triangle (one with 2 equal sides, 2 equal angles), and a parallelogram:

But she could not make an equilateral triangle (one with all equal sides, all equal angles).