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MEASUREMENT OF ANGLES

Definition of an Angle

An angle is formed when two straight lines meet at a point. The point where they meet is called the vertex, and the lines are called the arms of the angle.

The size of an angle is measured in degrees (°).

Example:

   B
   |
   |
   A
  /
 /
C

Here, ∠BAC is an angle with vertex at A and arms AB and AC.

Types of Angles
  • Acute angle: Less than 90°

  • Right angle: Exactly 90°

  • Obtuse angle: More than 90° but less than 180°

  • Straight angle: Exactly 180°

  • Reflex angle: More than 180° but less than 360°




Measurement of Angles
  • Use a protractor to measure angles.

  • Place the midpoint of the protractor at the vertex.

  • Align one arm of the angle with the zero line on the protractor.

  • Read the scale where the other arm crosses the protractor.




Special Angles and Their Properties

1. Vertically Opposite Angles

  • Formed when two lines intersect.

  • Opposite angles are equal.

   X
    \
     \
      • A
     /
    /
   Y

Here, ∠XAY = ∠YAZ

Property: Vertically opposite angles are always equal.

2. Adjacent Angles

  • Two angles that have a common vertex and a common arm and do not overlap.

   B
   |
   |
A--+--C

∠BAD and ∠DAC are adjacent.

Property: Sum of adjacent angles depends on the situation (if they form a straight line, sum = 180°).

3. Alternate Angles

  • Occur when a line (transversal) crosses two parallel lines.

  • Angles on opposite sides of the transversal but inside the parallel lines are alternate.

  • Alternate angles are equal.

L1: --------
       \
        \
L2: --------

4. Corresponding Angles

  • When a transversal cuts two parallel lines, angles in the same position at each intersection are corresponding.

  • Corresponding angles are equal.

L1: --------
       \
        \
L2: --------

5. Angles on a Straight Line

  • Sum of angles on a straight line = 180°

A--------B--------C

∠ABC + ∠CBD = 180°

6. Angles at a Point

  • Sum of angles around a point = 360°

     D
     |
     |
A----O----B
     |
     |
     C

∠AOB + ∠BOC + ∠COD + ∠DOA = 360°




Measurement of Angles — Questions and Solutions

Question 1 – Angles on a straight line

Two angles on a straight line are 70° and another unknown angle. Find the unknown angle.

Diagram:

A ----(70°)---- B ---- (?) ---- C

Solution:

Step 1: Angles on a straight line add up to 180°.

Step 2: So, 70° + ? = 180°

Step 3: ? = 180° – 70° = 110°

Answer: The unknown angle is 110°.

Question 2 – Angles at a point

Four angles meet at a point. Three of them are 90°, 85°, and 100°. Find the fourth angle.

Diagram:

     /
 85°/ 
   / O
90°\ 
    \100°

Solution:

Step 1: Angles around a point add up to 360°.

Step 2: Add the known angles: 90° + 85° + 100° = 275°.

Step 3: Subtract from 360°: 360° – 275° = 85°.

Answer: The fourth angle is 85°.

Question 3 – Vertically opposite angles

Two straight lines cross. One angle is 55°. Find the opposite angle.

Diagram:

   \ /
55°  X
   / \

Solution:

Step 1: Vertically opposite angles are equal.

Step 2: So the opposite angle = 55°.

Answer: The opposite angle is 55°.

Question 4 – Adjacent angles on a straight line

If one angle on a straight line is 150°, find the other.

Solution:

Step 1: Angles on a straight line add to 180°.

Step 2: 180° – 150° = 30°.

Answer: The other angle is 30°.

Question 5 – Corresponding angles

Two parallel lines are cut by a transversal. One angle at the top intersection is 75°. Find the corresponding angle.

Diagram:

Line 1:  -----------75°---
            \
             \
Line 2:  -----------?-----

Solution:

Step 1: Corresponding angles are equal when lines are parallel.

Step 2: So the corresponding angle = 75°.

Answer: The corresponding angle is 75°.

Question 6 – Alternate angles

Two parallel lines are cut by a transversal. One interior angle is 65°. Find the alternate interior angle.

Solution:

Step 1: Alternate interior angles are equal.

Step 2: So the alternate interior angle = 65°.

Answer: The alternate angle is 65°.

Question 7 – Angles in a triangle

A triangle has two angles of 40° and 70°. Find the third angle.

Diagram:

   /\
40°/  \70°
 /____\

Solution:

Step 1: Angles in a triangle add up to 180°.

Step 2: Add the known angles: 40° + 70° = 110°.

Step 3: Subtract from 180°: 180° – 110° = 70°.

Answer: The third angle is 70°.

Question 8 – Right triangle

One angle of a right triangle is 30°. Find the third angle.

Solution:

Step 1: In a right triangle, one angle = 90°.

Step 2: Angles in triangle = 180°.

Step 3: Add known angles: 90° + 30° = 120°.

Step 4: Subtract from 180°: 180° – 120° = 60°.

Answer: The third angle is 60°.

Question 9 – Quadrilateral

A quadrilateral has angles 90°, 80°, and 100°. Find the fourth angle.

Diagram:

 ____90° 
|     \
|      \
80°     100°

Solution:

Step 1: Angles in a quadrilateral add to 360°.

Step 2: Add given angles: 90° + 80° + 100° = 270°.

Step 3: Subtract from 360°: 360° – 270° = 90°.

Answer: The fourth angle is 90°.

Question 10 – Exterior angle of a triangle

One exterior angle of a triangle is 120°. Find the two opposite interior angles if they are equal.

Diagram:

     /\
    /  \
120°\__/

Solution:

Step 1: Exterior angle = sum of opposite interior angles.

Step 2: So 120° = a + a = 2a.

Step 3: Divide by 2 → a = 60°.

Answer: The two opposite interior angles are 60° each.




Measurement of Angles with algebra — Questions and Solutions

Question 1 — Vertically opposite with algebra

Question: Two straight lines cross at point O. One angle is 2x+10° and the angle opposite it is 3x−5°. Find x and the size of each angle. Then find the other two angles at the intersection.

Diagram:

    \       /
     \     /
      \   /
       \ /
    A --O-- B
       / \
      /   \
     /     \
    /       \
   D         C

(Angles at O are: ∠AOB = 2x+10, ∠COD = 3x−5 — these two are vertically opposite.)

Solution (step by step):

Vertically opposite angles are equal. So set them equal:

2x+10 = 3x−5

Solve for x:

Subtract 2x from both sides → 10 = x−5

Add 5 to both sides → x=15

Find the angle size: substitute x=15 into 2x+10:

2(15)+10 = 30+10 = 40°

So the opposite angle 3x−5 = 3(15)−5 = 45−5 = 40° (check OK).

The other two angles are adjacent to the 40° angles and lie on a straight line with them. Angles on a straight line add to 180°.

So each adjacent angle = 180°−40°=140°.

Answer: x=15. Two opposite angles = 40° and the other two = 140°.

Question 2 — Adjacent angles (linear pair) with algebra

Question: Two angles on a straight line are 3x and x+20. They form a straight line. Find x and the size of each angle.

Diagram:

P ---- Q ---- R
     <--->
     3x    x+20

Solution (step by step):

Angles on a straight line add to 180°. So:

3x+(x+20)=180

Combine like terms: 4x+20=180

Subtract 20: 4x=160

Divide by 4: x=40

Find angles: 3x=3(40)=120°. x+20=40+20=60°.

Check: 120°+60°=180° — OK.

Answer: x=40. The angles are 120° and 60°.

Question 3 — Alternate / corresponding angles (parallel lines reasoning)

Question: Two parallel lines l and m are cut by a transversal t. At the top intersection the angle on the upper right is 70°. Find:
a) the corresponding angle at the bottom intersection,
b) the alternate interior angle, and
c) the supplementary angle adjacent to the 70° (the one on the left of the top intersection).

Diagram:

   l:  -----------------
            \ 70°  |
             \     |
              \    |  t (transversal)
   m:  -----------------
            \  ?   |

Solution (step by step):

Corresponding angles: equal when lines are parallel. So corresponding angle=70°.

Alternate interior angle: also equal. So angle=70°.

Supplementary adjacent angle: 180°−70°=110°.

Answer: corresponding=70°; alternate interior=70°; adjacent=110°.

Question 4 — Angles at a point (expressions in ratio)

Question: Four angles around point O are in the ratio 1:2:3:4. Find the size of each angle.

Diagram:

     ↑
  4 |   1
←----O----→
  3 |   2
     ↓

Solution (step by step):

Sum of angles at a point=360°.

Let smallest angle=x. Then angles are x,2x,3x,4x.

x+2x+3x+4x=10x=360 → x=36°

Now angles: x=36°, 2x=72°, 3x=108°, 4x=144°

Check: 36+72+108+144=360 — OK.

Answer: 36°,72°,108°,144°

Question 5 — Triangle with algebra (angle sum)

Question: In triangle ABC the angles are: ∠A=2x, ∠B=3x+10, ∠C=x+20. Find x and the three angles.

Diagram:

   C
  / \
 /   \
A-----B

Solution (step by step):

Sum in triangle=180°: 2x+(3x+10)+(x+20)=180

Combine: 6x+30=180 → 6x=150 → x=25

∠A=2(25)=50°, ∠B=3(25)+10=85°, ∠C=25+20=45°

Check: 50+85+45=180 — OK.

Answer: x=25. Angles: 50°,85°,45°.

Question 6 — Parallel lines with algebraic angles

Question: Two parallel lines are cut by a transversal. At the top intersection the angle in one position is 4x−10°. At the bottom intersection the corresponding angle is 2x+50°. Find x and the common angle. Then find the supplementary angle next to it.

Diagram:

Top line:    -------------------------
                 \   (4x - 10)°
                  \
                   \
Bottom line: -------------------------
                 \
                  \
                   \  (2x + 50)°

Solution (step by step):

Corresponding angles are equal: 4x−10=2x+50

Solve: 2x−10=50 → 2x=60 → x=30

Angle=4(30)−10=110° (check with other: 2(30)+50=110°)

Supplementary angle=180−110=70°

Answer: x=30. Common angle=110°, supplementary=70°.




Conclusion

Measurement of angles helps us to know the size of turns and corners in shapes.

Angles can be on a straight line, around a point, inside triangles or other shapes.

Knowing these rules makes solving geometry questions easier.




CHECK OTHER RELATED TOPICS HERE


  1. PLANE SHAPE

  2. PERIMETER OF REGULAR POLYGONS


  3. AREA OF REGULAR PLANE SHAPES

  4. THREE-DIMENSIONAL FIGURES


  5. CONSTRUCTION

  6. MEASUREMENT OF ANGLES


  7. STATISTICS
    8.



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