Triangles - Olympiad Level MCQ, Class 9 Mathematics - Class 9 MCQ

In triangle ABC, we know that the sum of angles in a triangle is 180°. Therefore, we can express the angle ∠C as follows:

• ∠C = 180° - (∠A + ∠B) = 65°

Using the Law of Sines, the sides are proportional to the sines of their opposite angles. Given the angles:

• ∠A = 45°
• ∠B = 70°
• ∠C = 65°

We can determine the relationships between the sides:

• Since ∠A (45°) < ∠C (65°) < ∠B (70°),
• the shortest side is BC (opposite ∠A).
• The largest side is AC (opposite ∠B).

Thus, the correct answer is option B: BC, AC.

Let’s denote the common value as k. Therefore, we have:

• 2
• 3
• 6

Since the sum of angles in a triangle is 180^°:

k/2 + k/3 + k/6 = 180.

Finding a common denominator (6):

• (3k/6) + (2k/6) + (k/6) = 180.

Simplifying:

• 6k/6 = 180 implies k = 180.

Thus, the angles are:

• °,
• °,
• °.

Therefore, the measures of the angles are 90^°, 60^°, and 30^°, corresponding to option B.

Triangles ADB and ADC are right-angled at D, sharing equal hypotenuses:

• AB = AC

This equality makes them congruent by the RHS property:

• R - Right angle at D
• H - Equal hypotenuses (AB = AC)
• S - One side (AD) is the same in both triangles

x = 180 - 100 [L.P. of angles] = 80°

To determine which set of angles can form a triangle, we check if the sum of each set equals 180°.

• Option A: 70 + 90 + 25 = 185° (Exceeds 180°)
• Option B: 65 + 85 + 40 = 190° (Exceeds 180°)
• Option C: 65 + 85 + 30 = 180° (Valid triangle)
• Option D: 45 + 45 + 80 = 170° (Less than 180°)

Only Option C adds up to exactly 180°, making it the correct answer.

In a triangle, according to the triangle inequality theorem, the sum of the lengths of any two sides must be greater than the length of the remaining side. This ensures that the three sides can form a valid triangle.

Therefore, the correct answer is 'Greater than.'

The sum of the angles in any triangle must be exactly 180°. Adding the given angles:

• 90°
• 90°
• 20°

This results in:

90° + 90° + 20° = 200°, which exceeds 180°. Therefore, they cannot form a valid triangle.