Dear Readers,
As you know lines and angles is critical in the study of geometry. Whether we know it or not, we see all sorts of angles on an everyday basis. In order to continue our study of geometry, it will be necessary to learn about the different properties of lines, angles, and their measures. The sections below introduce the different classifications of angles (acute, obtuse, right, or straight).
Right angle: An angle of 90° is called a right angle.
Acute angle: An angle of less than 90° is called an acute angle.
Obtuse angle: An angle of greater than 90° is called an obtuse angle.
Reflex angle: An angle of greater than 180° is called reflex angle.
Complementary angles: If the sum of the two angles is 90°, then they are called complementary angles.
Supplementary angles: If the sum of the two angles is 180°, then they are called supplementary angles.
Acute angle: An angle of less than 90° is called an acute angle.
Obtuse angle: An angle of greater than 90° is called an obtuse angle.
Reflex angle: An angle of greater than 180° is called reflex angle.
Complementary angles: If the sum of the two angles is 90°, then they are called complementary angles.
Supplementary angles: If the sum of the two angles is 180°, then they are called supplementary angles.
Angle Relations: Vertically opposite angles: If two lines intersect each other, then the vertically opposite angles are equal. ∠a = ∠c and ∠b = ∠d
a, c and b, d are vertically opposite angles.
a, c and b, d are vertically opposite angles.
Angles in parallel lines: If a transversal intersects two parallel lines:
a. each pair of consecutive interior angles are supplementary.
∠4 + ∠5 = 180°
∠3 + ∠6 = 180°
b. each pair of alternate interior angles are equal.
∠5 =∠3, ∠6 = ∠4
∠5 = ∠3 =∠1 =∠7
∠6 = ∠4 = ∠2 = ∠8
∠4 + ∠5 = 180°
∠3 + ∠6 = 180°
b. each pair of alternate interior angles are equal.
∠5 =∠3, ∠6 = ∠4
∠5 = ∠3 =∠1 =∠7
∠6 = ∠4 = ∠2 = ∠8
Collinear: Three or more than three points are said to be collinear, if there is a line which contains them all.
Concurrent: Three or more than three lines are said to be concurrent, if there is a point which lies on them all.
Proportionality Theorem:
In the above diagram, ABBC=DEEF or ACBC=DFEF
Quiz on above topics :
1. In the adjoining figure, find ∠ C. If AB // CD
1. In the adjoining figure, find ∠ C. If AB // CD
(a) 120
(b) 50
(c) 110
(d) 100
2. Find the value of x in following figure.
(a) 110
(b) 100
(c) 130
(d) None of these
3. Find ∠OBD in the given figure. It is given that EC || AO.
(a) 20
(b) 50
(c) 40
(d) None of these
4. If the two lines n and m are parallel, then find the value of ϕ in the given figure.
(a) 40
(b) 80
(c) 30
(d) None of these
Answers With Solutions:
1. (c)
Sol: As C // E, ∠B = ∠Y
Answers With Solutions:
1. (c)
∠ E = 180 - 120 = 60
∠ E + ∠ X + ∠ F = 180 ⇒ 60 + 50 + F = 180 ⇒ F = 70
Now ∠ G = 180 - F = 110.
As AB // CD, ∠G = ∠C
So ∠C = 110
2. (a) Now ∠ G = 180 - F = 110.
As AB // CD, ∠G = ∠C
So ∠C = 110
Sol: As C // E, ∠B = ∠Y
Also ∠Y + ∠Z = 180 ⇒ ∠Z = 180 - 130 = 50
Now in Δ AGF, ∠A + ∠Z + ∠X = 180 ⇒ ∠X = 180 - 20 - 50 = 110
3. (b)
Sol: From the diagram, in ΔOBD, 110 + 20 + ∠B = 180 ⇒ ∠B = 50
4.(b)
Sol: Draw a parellel to n. Now from the diagram ϕ = 50 + 30 = 80 (Alternate interior angles are equal)
