Angles

An angle is named by taking a point on one arm, then the vertex, and then a point on the other arm. The common point of the two arms is the vertex.

(i) The angle is ∠ACB (or ∠BCA); its vertex is C and its arms are $\mathbf{\overrightarrow{CA} \text{ and } \overrightarrow{CB}}$.

(ii) The angle is ∠PTM (or ∠MTP); its vertex is T and its arms are $\mathbf{\overrightarrow{TP} \text{ and } \overrightarrow{TM}}$.

(iii) The angle is ∠LSR (or ∠RSL); its vertex is S and its arms are $\mathbf{\overrightarrow{SL} \text{ and } \overrightarrow{SR}}$.

(iv) The angle is ∠XZY (or ∠YZX); its vertex is Z and its arms are $\mathbf{\overrightarrow{ZX} \text{ and } \overrightarrow{ZY}}$.

(v) The angle is ∠EDF (or ∠FDE); its vertex is D and its arms are $\mathbf{\overrightarrow{DE} \text{ and } \overrightarrow{DF}}$.

(vi) The angle is ∠HKG (or ∠GKH); its vertex is K and its arms are $\mathbf{\overrightarrow{KH} \text{ and } \overrightarrow{KG}}$.

(i) It is a triangle PQR, so three angles are formed, namely ∠PQR, ∠QRP and ∠QPR.

(ii) It is a quadrilateral ABCD, so four angles are formed, namely ∠ABC, ∠BCD, ∠CDA and ∠DAB.

(iii) In this figure (a quadrilateral EFGH with a diagonal), eight angles are formed, namely ∠HEF, ∠EFH, ∠EHF, ∠EHG, ∠EFG, ∠HFG, ∠FHG and ∠HGF.

The rays OA, OB, OC and OD start from the common point O. Taking the arms two at a time, the possible angles are:

∠AOB, ∠AOC, ∠AOD, ∠BOC, ∠BOD and ∠COD.

Naming each angle by a point on one arm, the vertex O, and a point on the other arm:

⇒ ∠a = ∠AOB

⇒ ∠b = ∠BOP

⇒ ∠c = ∠POQ

⇒ ∠1 = ∠QOR

⇒ ∠2 = ∠ROA

(i) The points lying on the angle (on its arms or at the vertex) are Y, U, O, P, B and X.

(ii) The points lying in the interior of the angle are T, A, S and Z.

(iii) The points lying in the exterior of the angle are C, R, D and L.

(i) False. Point C lies on the arm OC of ∠AOC, so it lies on the angle, not in its interior.

(ii) True. The ray OC lies between OA and OD, so point C is in the interior of ∠AOD.

(iii) False. The ray OD lies outside ∠AOC, so point D is not in the interior of ∠AOC.

(iv) True. The ray OB lies outside ∠AOD, so point B is in the exterior of ∠AOD.

(v) False. Point C lies in the interior of ∠AOB, not on it.

To measure each angle, place the protractor so that its centre is exactly on the vertex and its base line lies along one arm. Then read off the mark on the scale (starting from 0°) through which the other arm passes.

(i) 50°

(ii) 110°

(iii) 50°

(iv) 80°

(v) 80°

(vi) 110°

To measure each angle, place the protractor so that its centre is exactly on the vertex and its base line lies along one arm. Then read off the mark on the scale (starting from 0°) through which the other arm passes.

The angle is then named according to its measure — acute (more than 0° but less than 90°), right (exactly 90°), obtuse (more than 90° but less than 180°), straight (exactly 180°) or reflex (more than 180° but less than 360°).

(i) The arm passes through the 90° mark, and an angle of exactly 90° is a right angle.

90°, a right angle.

(ii) The arm passes through the 115° mark. Since 90° < 115° < 180°, it is an obtuse angle.

115°, an obtuse angle.

(iii) The two arms lie in opposite directions along the same straight line and the arm passes through the 180° mark, so it is a straight angle.

180°, a straight angle.

(iv) The arm passes through the 145° mark. Since 90° < 145° < 180°, it is an obtuse angle.

145°, an obtuse angle.

(v) The arm passes through the 78° mark. Since 0° < 78° < 90°, it is an acute angle.

78°, an acute angle.

(vi) The arm passes through the 120° mark. Since 90° < 120° < 180°, it is an obtuse angle.

120°, an obtuse angle.

An angle whose measure is more than 90° but less than 180° is called an obtuse angle. Since 90° < 93° < 180°, the given angle is an obtuse angle.

Hence, 93° is an obtuse angle.

An angle whose measure is exactly 180° is called a straight angle. The given angle measures 180°, so it is a straight angle.

Hence, 180° is a straight angle.

An angle whose measure is more than 0° but less than 90° is called an acute angle. Since 0° < 86° < 90°, the given angle is an acute angle.

Hence, 86° is an acute angle.

An angle whose measure is more than 90° but less than 180° is called an obtuse angle. Since 90° < 125° < 180°, the given angle is an obtuse angle.

Hence, 125° is an obtuse angle.

An angle whose measure is 0° is called a zero angle. It is formed when the two arms of the angle coincide with each other.

Hence, 0° is a zero angle.

An angle whose measure is more than 90° but less than 180° is called an obtuse angle. Since 90° < 90.5° < 180°, the given angle is an obtuse angle.

Hence, 90.5° is an obtuse angle.

An angle whose measure is more than 0° but less than 90° is called an acute angle. Since 0° < 89.5° < 90°, the given angle is an acute angle.

Hence, 89.5° is an acute angle.

An angle whose measure is exactly 90° is called a right angle. The given angle measures 90°, so it is a right angle.

Hence, 90° is a right angle.

An angle whose measure is 360° is called a complete angle. It is formed when a ray makes one full rotation about its end point and comes back to its initial position.

Hence, 360° is a complete angle.

An angle whose measure is more than 180° but less than 360° is called a reflex angle. Since 180° < 182° < 360°, the given angle is a reflex angle.

Hence, 182° is a reflex angle.

An angle whose measure is more than 0° but less than 90° is called an acute angle. Since 0° < 3° < 90°, the given angle is an acute angle.

Hence, 3° is an acute angle.

An angle whose measure is more than 0° but less than 90° is called an acute angle. Since 0° < 25° < 90°, the given angle is an acute angle.

Hence, 25° is an acute angle.

We know that 1 right angle = 90°.

One right angle = 90°.

Hence, one right angle = 90°.

We know that 1 right angle = 90°.

Two right angles = 2 × 90° = 180°.

Hence, two right angles = 180°.

We know that 1 right angle = 90°.

Three right angles = 3 × 90° = 270°.

Hence, three right angles = 270°.

We know that 1 right angle = 90°.

$\frac{3}{5}$ right angles = $\frac{3}{5}$ × 90° = 54°.

Hence, $\mathbf{\frac{3}{5}}$ right angles = 54°.

We know that 1 right angle = 90°.

$\frac{3}{2}$ right angles = $\frac{3}{2}$ × 90° = 135°.

Hence, $\mathbf{\frac{3}{2}}$ right angles = 135°.

We know that 1 right angle = 90°.

Four right angles = 4 × 90° = 360°.

Hence, four right angles = 360°.

An angle whose measure is more than 180° but less than 360° is a reflex angle. Since 180° < 210° < 360°, the angle is a reflex angle.

Hence, option 3 is the correct option.

An angle whose measure is more than 180° but less than 360° is a reflex angle. Since 180° < 340° < 360°, the angle is a reflex angle.

Hence, option 2 is the correct option.

An angle whose measure is more than 90° but less than 180° is an obtuse angle. Since 90° < 92° < 180°, the angle is an obtuse angle.

Hence, option 3 is the correct option.

An angle whose measure is 180° is a straight angle.

Hence, option 1 is the correct option.

An angle whose measure is 360° is a complete angle.

Hence, option 2 is the correct option.

We know that 1 complete angle = 360°.

⇒ $\frac{5}{6}$ of a complete angle = $\frac{5}{6}$ × 360° = 300°.

Hence, option 3 is the correct option.

A reflex angle is an angle whose measure is more than 180° but less than 360°.

Hence, option 3 is the correct option.

The measure of a straight angle is 180°.

Hence, option 2 is the correct option.

(i) 180°. A straight angle is formed when the two arms of the angle lie in opposite directions along the same straight line. Half of a complete rotation is $\frac{1}{2}$ × 360° = 180°.

(ii) 360°. A complete angle is formed when a ray rotates about its end point and comes back to its initial position, that is, it makes one full rotation. Such a rotation measures 360°.

(iii) 180° and 360°. An angle is called a reflex angle when it is bigger than a straight angle but smaller than a complete angle, so its measure lies between 180° and 360°.

(iv) 90° and 180°. An angle is called an obtuse angle when it is bigger than a right angle but smaller than a straight angle, so its measure lies between 90° and 180°.

(v) vertex. The two rays forming the angle are called its arms or sides, and the common end point at which the two arms meet is called the vertex of the angle.

(i) True. An angle whose measure is 180° is a straight angle.

(ii) True. An angle whose measure is 360° is a complete angle.

(iii) False. An angle measuring 180° is a straight angle; an obtuse angle lies between 90° and 180°.

(iv) False. An acute angle lies between 0° and 90°. Angles between 0° and 180° also include right and obtuse angles.

(v) True. $\frac{2}{3}$ of a straight angle = $\frac{2}{3}$ × 180° = 120°.

Assertion (A) is true, because a horizontal line and a vertical line are perpendicular to each other and hence always intersect at right angles.

Reason (R) is true, because the measure of a right angle is indeed 90°.

Reason (R) only tells us the measure of a right angle. It does not explain why a horizontal line and a vertical line intersect at right angles. So Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.

Assertion (A) is true, because an angle whose measure is 360° is called a complete angle.

Reason (R) is true, because the total angle at a point is 360°.

Since the angle at a point is 360°, a ray making one complete rotation about that point traces an angle of 360°, and this is exactly why such an angle is called a complete angle. So Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

From the figure, D, O and C lie on a straight line, so ∠DOC = 180°. Also ∠AOD = 90° and ∠BOC = 145°.

Since ∠BOD and ∠BOC lie on the straight line DOC,

⇒ ∠BOD + ∠BOC = 180°

⇒ ∠BOD = 180° − 145° = 35°

The ray OB lies between OD and OA, so

⇒ ∠AOB = ∠AOD − ∠BOD = 90° − 35° = 55°.

So, ∠BOD = 35° and ∠AOB = 55°.

Hence, option 2 is the correct option.

From the figure, the ray OC lies between OA and OB, so ∠AOB is divided into ∠AOC = (4x − 7)° and ∠BOC = (8x + 1)°.

Since ∠AOC + ∠BOC = ∠AOB = 90°,

⇒ (4x − 7)° + (8x + 1)° = 90°

⇒ 4x° - 7° + 8x° + 1° = 90°

⇒ 12x° − 6° = 90°

⇒ 12x° = 96°

⇒ x° = 8°

⇒ x = 8.

Hence, option 2 is the correct option.

The rays drawn from point O divide the full angle around O into several angles. Since the angles around a point add up to 360°, their sum is 360°.

⇒ (2x − 40)° + (3x)° + (5x + 15)° + (3x − 30)° + (2x − 35)° = 360°

⇒ 2x° - 40° + 3x° + 5x° + 15° + 3x° - 30° + 2x° - 35° = 360°

⇒ 15x° − 90° = 360°

⇒ 15x° = 450°

⇒ x° = 30°

⇒ x = 30.

Hence, option 4 is the correct option.

Turning clockwise from South, we pass through:

South → West → North → East

That is three turns of 90° each (South to West, West to North, North to East), i.e. three right angles.

Hence, option 2 is the correct option.

In the word GREAT, the letters made up of lines meeting at right angles (perpendicular lines) are E and T. So there are 2 such letters.

Hence, option 1 is the correct option.

From the figure:

⇒ ∠AOB is less than 90°, so it is an acute angle → (ii)

⇒ ∠AOE measures 90°, so it is a right angle → (iii)

⇒ ∠AOC is more than 90° but less than 180°, so it is an obtuse angle → (iv)

⇒ ∠BOD measures 180°, so it is a straight angle → (i)

Thus, the correct matching is:

Column I	Column II
(A) ∠AOB	(ii) Acute angle
(B) ∠AOE	(iii) Right angle
(C) ∠AOC	(iv) Obtuse angle
(D) ∠BOD	(i) Straight angle

Hence, option 3 is the correct option.

Statement–I: In the figure, A, F and E lie on a straight line and the rays FB, FC and FD are drawn from F so that

⇒ ∠AFB = ∠BFC = ∠CFD = ∠DFE = 45°

Taking these rays two at a time, the angles formed at F are:

⇒ ∠AFB = 45°, ∠BFC = 45°, ∠CFD = 45° and ∠DFE = 45° — each an acute angle

⇒ ∠AFC = 45° + 45° = 90°, ∠BFD = 45° + 45° = 90° and ∠CFE = 45° + 45° = 90° — each a right angle

⇒ ∠AFD = 45° + 45° + 45° = 135° and ∠BFE = 45° + 45° + 45° = 135° — each an obtuse angle

⇒ ∠AFE = 45° + 45° + 45° + 45° = 180° — a straight angle

So the angles measuring 135° are ∠AFD (made up of ∠AFB, ∠BFC and ∠CFD) and ∠BFE (made up of ∠BFC, ∠CFD and ∠DFE). Since an obtuse angle is more than 90° but less than 180°, only these two angles are obtuse.

∴ The number of obtuse angles is 2, not 4. So Statement–I is false.

Statement–II: If the angle between two intersecting lines is a right angle, the lines are perpendicular, not parallel. So Statement–II is false.

Hence, option 2 is the correct option.

One revolution (complete angle) = 360°.

Option 1: A right angle = 90° = $\frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}$ of a revolution, not $\frac{1}{2}$.

So option 1 is false.

Option 2: A straight angle = 180° = $\frac{180^{\circ}}{360^{\circ}} = \frac{1}{2}$ of a revolution, not $\frac{1}{4}$.

So option 2 is false.

Option 3: A reflex angle is more than 180° and a straight angle is exactly 180°, so a reflex angle is greater than a straight angle.

So option 3 is correct.

Option 4: The angle formed in one revolution is 360°, which is a complete angle, not a reflex angle.

So option 4 is false.

Hence, option 3 is the correct option.