Olympiad Test: Lines and Angles - Class 7 Maths Free MCQ with solutions

∵ In ∆ABC,
AB = AC
∴ ∠B = ∠C    ...(1)
| Angles opposite to equal sides of a triangle are equal
In ∆ABC,
∠A + ∠B + ∠C = 180°
| Sum of all the angles of a triangle is 180°
⇒ 90° + ∠B + ∠C = 180°
| ∵ ∠A = 90° (given)
⇒    ∠B + ∠C = 90°    ...(2)
From (1) and (2), we get
∠B = ∠C = 45°.

Alternate exterior angles are pairs of angles that are located outside two parallel lines and on opposite sides of a transversal. In the given figure, the pairs of alternate exterior angles can be identified as follows:

• Angle 1 and Angle 4 are alternate exterior angles.
• They are on opposite sides of the transversal.
• They lie outside the parallel lines.

In contrast, other pairs mentioned do not meet these criteria:

• Angle 1 and Angle 2 are not alternate exterior angles as they are on the same side of the transversal.
• Angle 1 and Angle 5 are also on the same side of the transversal.

Thus, the correct pair of alternate exterior angles is Angle 1 and Angle 4.

Supplementary angles sums up to 180°
x + 105° = 180°
x = 180° - 105°
x = 75°

Complementary angle sums up to 90°. So x+ 45° = 90°
x = 90° - 45° = 45°

Let's find the complement of 65°

Complement of an angle is the angle that when added to it gives 90°.

So, if x is the complement of 65°, then:

• x + 65° = 90°

To find x, subtract 65° from both sides:

• x = 90° - 65°
• x = 25°

Therefore, the complement of 65° is 25°.

Let the angle be x°
then,complement°=(90-x)°

A/Q,
x=(90-41)
x=90-41
x=49

As we know for becoming supplementary angles, the sum of both the angles should be 180^∘
Let the supplementary angle be x.
So, 54^∘+x = 180^∘
⇒ x = 126^∘

To determine which pair of angles is supplementary, we check if their sum equals 180°.

• Option A: 45° + 45° = 90° (Not supplementary)
• Option B: 63° + 27° = 90° (Not supplementary)
• Option C: 112° + 68° = 180° (Supplementary)
• Option D: 80° + 10° = 90° (Not supplementary)

Thus, the correct answer is option C.

Step 1: Recall the definition of complementary angles
Two angles are complementary if their sum is 90°. Let the angle be xxx. Then its complement is 90−x90 - x90−x.

Step 2: Set up the equation
If the angle is equal to its complement:
x = 90 - x

Step 3: Solve for x
x + x = 90
2x = 90
x = 45

Step 4: Verify
Complement of 45° = 90 - 45 = 45°

Supplementary angles sums up to 180° since 90° + 90° = 180°, 90° is supplementary to itself

120 and x are corresponding angles and corresponding angles are equal so x = 120^°

The angles are co-interior so x+60 = 180
x=120^°

as we know alternate angles are equal x = 60^°

Given,
∠A = 90° and AB = AC
A/q,
AB = AC 
⇒ ∠B = ∠C (Angles opposite to the equal sides are equal.)
Now,
∠A + ∠B + ∠C = 180° (Sum of the interior angles of the triangle.)
⇒ 90° + 2∠B = 180°
⇒ 2∠B = 90° 
⇒ ∠B = 45°
Thus, ∠B = ∠C = 45°

To find the supplementary angle, we know that the sum of an angle and its supplement is 180°. Therefore, if we let x be the measure of the supplementary angle:

• x + 70° = 180°

Solving for x gives:

• x = 110°

This will be the equation x=5(90-x)
solution= x=5(90-x)              
                x+5x=450
                6x=450
                 6x/6=450/6   {divide by 6 to both the side to x}
                  x=75

ans= The angle is 75 degree and its compliment angle is 15 degree.

x + 2x = 180 3x= 180 x=60

c + 110^ο = 180^ο
c = 70^ο
x = c [ alternate interior angle ]
so x =70^ο​​​​​