Fundamental concepts of Geometry:
Point: It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e., it has no magnitude.
Line segment: The straight path joining two points A and B is called a line segment AB . It has and points and a definite length.
Ray: A line segment which can be extended in only one direction is called a ray.
Intersecting lines: Two lines having a common point are called intersecting lines. The common point is known as the point of intersection.
Concurrent lines: If two or more lines intersect at the same point, then they are known as concurrent lines.
Angles: When two straight lines meet at a point they form an angle.
Right angle: An angle whose measure is 90° is called a right angle.
Acute angle: An angle whose measure is less then one right angle (i.e., less than 90°), is called an acute angle.
Obtuse angle: An angle whose measure is more than one right angle and less than two right angles (i.e., less than 180° and more than 90°) is called an obtuse angle.
Reflex angle: An angle whose measure is more than 180° and less than 360° is called a reflex angle.
Complementary angles: If the sum of the two angles is one right angle (i.e.,90°), they are called complementary angles. Therefore, the complement of an angle θ is equal to 90° - θ.
i.e.∠A + ∠B = 90° or ∠A = ∠90°-∠B - complement of ∠A
Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180°
Example: Angles measuring 130° and 50° are supplementary angles. Two supplementary angles are the supplement of each other. Therefore, the supplement of an angle θ. is equal to 180° - θ..
Vertically opposite angles: When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles.
∠1,∠3 and ∠4,∠2 are vertically opposite angles
Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle.
Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side.
Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points
Angles formed by a transversal of two parallel lines:In the above figure, l and m are two parallel lines intersected by a transversal PS.
The following
properties of the angles can be observed:
∠3 = ∠5 and ∠4 = ∠6[Alternate angles]
∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, ∠3 = ∠7 [Corresponding angles]
∠4 + ∠5 = ∠3 + ∠6 = 180° [Supplementary angles]
Some Questions based on Concept:
1.In the given figure, the arms of two angles are parallel. If ∠ABC = 70° then find : ∠DGC and ∠DEF
(a) 70°, 60°
(b) 60°, 50°
(c) 70°, 70°
(d) None of these
2.Find y, if x° = 36° , as per the given diagram :
(a) 36°
(b) 16°
(c) 12°
(d) 42°
3.In figure, AB II CD, Find θ∶
(a) 30°
(b) 35°
(c) 40°
(d) 45°
4.In the figure, OP ⊥ OA and OQ ⊥ OB. Find ∠POQ if ∠AOB = 20°
(a) 20°
(b) 30°
(c) 40°
(d) None of these
5.In the figure , if PQ II ST, ∠PQR = 110° and ∠RST = 130° , find ∠QRS.
(a) 40°
(b) 50°
(c) 60°
(d) 70°
6.In the figure, AB IICD, the value of x is:
(a) 220°
(b) 140°
(c) 150°
(d) none of these
7.If a straight line L makes an angle θ (θ>90°) with the positive direction of x - axis, then the acute angle made by a straight line L1, perpendicular to L, with the y-axis is:
(a) π/2+ θ
(b) π/2- θ
(c) π+ θ
(d) π- θ
8.From the following figure, find ∠RQZ, if ∠RQZ = 2 ∠QRS and PQ II ST:
(a) 20°
(b) 30°
(c) 40°
(d) 60°
9.In the given figure, PR II TS and PU II RS. Find ∠TPU:
(a) 60°
(b) 70°
(c) 80°
(d) 100°
10.In the given figure, ∠a is greater than one-sixth of right angle, then:
(a) b > 165°
(b) b < 165°
(c) b ≤165°
(d) b ≥ 165°
ANSWERS AND SOLUTION:
1.(c) ∠DGC = ∠DEF = ∠ ABC = 70 (Corresponding angles)
2.(a) 2y + 3x = 180° ⇒y=36°
(∵x=36°)
or x + 4y =180°=> y = 36°
3.(C) ∠COE= ∠ABE=75°
(corresponding angles)
so ∠DOE=180°- ∠ COE=180°-75°=105°
in ∆DOE, 105° +35° + θ = 180°
θ = 180° - 140° = 40°
4.(a) ∠BOP=90°- ∠AOB
= 90° - 70°
= 20°
so ∠POQ=90°- ∠BOP
= 90° - 70°
- 20°
5.(c) draw a line in that QI, in that figure then
∠RQO=180°-110°=70°
∠ROI=130° (Corresponding)
So, ∠ROQ=180°-130°=50°
Now in ∆QRO
⇒∠RQO+ ∠ROQ+ ∠QRO=180°
⇒∠QRO=180°-(70°+50°)
= 60°=∠QRS
6.(b)
7.(d)
8.(c)
9.(b) PR II TS
∴ ∠PRQ= ∠USR=50°
In PQR :
∠PQR=180°-(50°+60°)
= 70°
∴ ∠TPU= ∠PQR =70°
[∴PU ΙΙ RS ΙΙ QS]
10.(b)
a > (90°)/6 ⇒a >15°
a + b = 180° ⇒b<165° (∵a<15°)
Point: It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e., it has no magnitude.
Line segment: The straight path joining two points A and B is called a line segment AB . It has and points and a definite length.
Ray: A line segment which can be extended in only one direction is called a ray.
Intersecting lines: Two lines having a common point are called intersecting lines. The common point is known as the point of intersection.
Concurrent lines: If two or more lines intersect at the same point, then they are known as concurrent lines.
Angles: When two straight lines meet at a point they form an angle.
Right angle: An angle whose measure is 90° is called a right angle.
Acute angle: An angle whose measure is less then one right angle (i.e., less than 90°), is called an acute angle.
Obtuse angle: An angle whose measure is more than one right angle and less than two right angles (i.e., less than 180° and more than 90°) is called an obtuse angle.
Reflex angle: An angle whose measure is more than 180° and less than 360° is called a reflex angle.
Complementary angles: If the sum of the two angles is one right angle (i.e.,90°), they are called complementary angles. Therefore, the complement of an angle θ is equal to 90° - θ.
i.e.∠A + ∠B = 90° or ∠A = ∠90°-∠B - complement of ∠A
Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180°
Example: Angles measuring 130° and 50° are supplementary angles. Two supplementary angles are the supplement of each other. Therefore, the supplement of an angle θ. is equal to 180° - θ..
Vertically opposite angles: When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles.
∠1,∠3 and ∠4,∠2 are vertically opposite angles
Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle.
Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side.
Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points
Angles formed by a transversal of two parallel lines:In the above figure, l and m are two parallel lines intersected by a transversal PS.
The following
properties of the angles can be observed:
∠3 = ∠5 and ∠4 = ∠6[Alternate angles]
∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, ∠3 = ∠7 [Corresponding angles]
∠4 + ∠5 = ∠3 + ∠6 = 180° [Supplementary angles]
Some Questions based on Concept:
1.In the given figure, the arms of two angles are parallel. If ∠ABC = 70° then find : ∠DGC and ∠DEF
(a) 70°, 60°
(b) 60°, 50°
(c) 70°, 70°
(d) None of these
2.Find y, if x° = 36° , as per the given diagram :
(a) 36°
(b) 16°
(c) 12°
(d) 42°
3.In figure, AB II CD, Find θ∶
(a) 30°
(b) 35°
(c) 40°
(d) 45°
4.In the figure, OP ⊥ OA and OQ ⊥ OB. Find ∠POQ if ∠AOB = 20°
(a) 20°
(b) 30°
(c) 40°
(d) None of these
5.In the figure , if PQ II ST, ∠PQR = 110° and ∠RST = 130° , find ∠QRS.
(a) 40°
(b) 50°
(c) 60°
(d) 70°
6.In the figure, AB IICD, the value of x is:
(a) 220°
(b) 140°
(c) 150°
(d) none of these
7.If a straight line L makes an angle θ (θ>90°) with the positive direction of x - axis, then the acute angle made by a straight line L1, perpendicular to L, with the y-axis is:
(b) π/2- θ
(c) π+ θ
(d) π- θ
8.From the following figure, find ∠RQZ, if ∠RQZ = 2 ∠QRS and PQ II ST:
(a) 20°
(b) 30°
(c) 40°
(d) 60°
9.In the given figure, PR II TS and PU II RS. Find ∠TPU:
(a) 60°
(b) 70°
(c) 80°
(d) 100°
10.In the given figure, ∠a is greater than one-sixth of right angle, then:
(a) b > 165°
(b) b < 165°
(c) b ≤165°
(d) b ≥ 165°
ANSWERS AND SOLUTION:
1.(c) ∠DGC = ∠DEF = ∠ ABC = 70 (Corresponding angles)
2.(a) 2y + 3x = 180° ⇒y=36°
(∵x=36°)
or x + 4y =180°=> y = 36°
3.(C) ∠COE= ∠ABE=75°
(corresponding angles)
so ∠DOE=180°- ∠ COE=180°-75°=105°
in ∆DOE, 105° +35° + θ = 180°
θ = 180° - 140° = 40°
4.(a) ∠BOP=90°- ∠AOB
= 90° - 70°
= 20°
so ∠POQ=90°- ∠BOP
= 90° - 70°
- 20°
5.(c) draw a line in that QI, in that figure then
∠RQO=180°-110°=70°
∠ROI=130° (Corresponding)
So, ∠ROQ=180°-130°=50°
Now in ∆QRO
⇒∠RQO+ ∠ROQ+ ∠QRO=180°
⇒∠QRO=180°-(70°+50°)
= 60°=∠QRS
6.(b)
7.(d)
8.(c)
9.(b) PR II TS
∴ ∠PRQ= ∠USR=50°
In PQR :
∠PQR=180°-(50°+60°)
= 70°
∴ ∠TPU= ∠PQR =70°
[∴PU ΙΙ RS ΙΙ QS]
10.(b)
a > (90°)/6 ⇒a >15°
a + b = 180° ⇒b<165° (∵a<15°)
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