Olympiad Mathematics - Lines and Angles
Exam Duration: 45 Mins Total Questions : 30
What do we call an angle which exactly measures 90°?
- (a)
An obtuse angle
- (b)
An acute angle
- (c)
A right angle
- (d)
A reflex angle
An angle which exactly measures 90° is called a right angle.
What do we call a 169° angle?
- (a)
An obtuse angle
- (b)
An acute angle
- (c)
A right angle
- (d)
A zero angle
An angle which lies between 90° and 180° is called as an obtuse angle. So, 169° is an obtuse angle.
What happens to the measurment of an angle after the extension of its arms?
- (a)
Doubles
- (b)
Triples
- (c)
Remains the same
- (d)
Cannot be said
Extending the arms of an angle does not affect the angle between them.
In \(\angle ROP\) what is the vertex?
- (a)
R
- (b)
P
- (c)
O
- (d)
PR
The vertex of an angle is the common point of the rays that form the arms of an angle. Here, it is O.
\(\overset { \longleftrightarrow }{ OQ } \bot \overset { \longleftrightarrow }{ PR } \) What is the measure of \(\angle QOR?\)
- (a)
1800
- (b)
450
- (c)
900
- (d)
1200
Which of the following is true?
- (a)
Two acute angles are supplementary
- (b)
Two obtuse angles are supplementary
- (c)
Two right angles are supplementary
- (d)
Two reflex angles are supplementary
Find the angle which is a complement of it self.
- (a)
30°
- (b)
45°
- (c)
90°
- (d)
180°
Observe the given figure in which l II m and n is the transversal and answer the questions that follow.
What type of angles are 'c' and 'p'?
- (a)
Corresponding angles
- (b)
Alternate angles
- (c)
Vertically opposite angles
- (d)
Interior angles on the same side of the transversal
l || m, n is the transversal. 'c' and 'p' are both interior angles, but on different sides of n. So, 'c' and 'p' are alternate angles.
In the given figure, what is the measure of x?
- (a)
32°
- (b)
148°
- (c)
64°
- (d)
180°
PR is a straight line and
so, x = 180° - 32° = 148°
In the figure \(\overline { AB } ,\overline { CD } and\quad \overline { EF } \) are three straight lines that interesect at O. If Y is thrice x, find the value of y.
- (a)
97.5°
- (b)
35°
- (c)
32.5°
- (d)
98°
From the figure
\(\angle AOC=50^0\)
(Vertically opposite angles)
Given y is thrice x, we have x + 50° +y = 180° (Angle on a striaght line)
\(\Rightarrow\) x+50°+3x = 180°
\(\Rightarrow x=\frac{130^0}{4}=32.5^0\)
\(\therefore\) y = 3x = 3(32.5) = 97.5°
In the figure, AB II CD and XY is the transversal.
Which of the following is incorrect?
- (a)
p = 115°
- (b)
q = 115°
- (c)
q = 65°
- (d)
r = 115°
115° and q are interior angles on the same side of the transversal.
So, 115°+q = 180°
\(\Rightarrow\) q = 180°-115° = 65°
Find the angle x in the given figure, if AB II CD.
- (a)
75°
- (b)
55°
- (c)
160°
- (d)
145°
Draw PQ II AB and CD
From the figure,
x = 20° + (180°- 55°) as
PQIIABIICD.
\(\Rightarrow\) x = 20° + 125° = 145°
Find the unknown angle x in the figure.
- (a)
45°
- (b)
125°
- (c)
90°
- (d)
80°
DFIICHIIBG
\(\Rightarrow\) a + b = x and a = 450 (Corresponding angles)
b = 180°-100° = 80°
(Angles on the same side of transversal.)
\(\Rightarrow\) x = a + b = 45° + 80° = 125°
Observe the figure given.
Compute the sum of x, y and z.
- (a)
180°
- (b)
70°
- (c)
190°
- (d)
80°
From the figure,
150 - x + 70 - x + x = 180°
\(\Rightarrow\) x = 220° -180° = 40°
Since AE II BD, y = x as they are alternate angles.
In \(\triangle BCD,\angle BDC=x\) (Alternate angles)
70 - x + x + z = 180°\(\Rightarrow\) z = 110°
\(\therefore\) The required sum = x+y+z = 40°+40°+ 110° = 190°
If \(\overrightarrow { OP } \) is a ray standing on a line \(\overset { \longleftrightarrow }{ QR } \) such that \(\angle POQ=\angle POR\) what is the measure \(\angle POQ?\)
- (a)
45°
- (b)
60°
- (c)
75°
- (d)
90°
Given OP is a rayon line QR.
Also \(\angle POQ=\angle POR\)
\(\angle POQ=\angle POR\).........(1)
\(\angle POQ+\angle POR=180^0\).........(2)
From (1) and (2), we have
\(2\angle POQ=180^0\)
\(\Rightarrow \angle POQ=\frac{180^0}{2}=90^0\)
In the figure, ACE, BCF and DCG are straight lines and AB II HC.
Find the angles p, q, r and s.
- (a)
p q r s 45° 50° 65° 50° - (b)
p q r s 45° 65° 50° 50° - (c)
p q r s 45° 65° 50° 65° - (d)
p q r s 45° 50° 65° 65°
In MBC, r = 180° - 50° - 65° = 65°
HC II AB \(\Rightarrow\) q = 50° . (Alternate angles) s = q (Vertically opposite angles)
Hence, s = 50°
Since BCF is a straight line,
p + 20° + q + r = 180° \(\Rightarrow\) p = 45°
\(\therefore\) p = 45°, q = 50°, r = 65° and s = 50° are the required values.
In the figure given, if AB II CD, what are the respective values of 'p' and 'q'?
- (a)
75° and 20°
- (b)
20° and 75°
- (c)
25° and 70°
- (d)
70° and 25°
The lines AB and EF intersect at G
\(\therefore \angle EGB=\angle AGF\)
(Vertically opposite angles)
\(\Rightarrow \angle AGF=65^0\)
Since AB||CD
\(\angle GHD=\angle AGH=\angle AGF\)
\(\Rightarrow \angle GHD=65^0\)
\(\Rightarrow GHO+\angle OHD=65^0\)
\(\Rightarrow\) q°=65°-40°=25°
Draw a line XY through 'O' parallel to AB and CD.
Since XY || AB, \(\angle XOG=\angle BGO\)
\(\Rightarrow \angle XOG=45^0\) (Alternate angles) and XY||CD \(\Rightarrow \angle XOH=\angle OHD\)
\(\Rightarrow \angle XOH=25^0\)
But \(p^0=\angle XOG+\angle XOH\)
\(\Rightarrow\) p = 45°+25° = 70°
\(\therefore\) P = 70° & q = 25°
In the figure given, a II band c II d. If \(\angle 1=75^0\) what is the measure of \(\angle 3?\)
- (a)
105°
- (b)
150°
- (c)
75°
- (d)
100°
Given a || b and c || d and \(\angle 1=75^0\)
Since a || b, \(\angle 1=\angle 2\)
Also c || d,\(\Rightarrow \angle 2+\angle 3=180^0\)
\(\therefore \angle 3=105^0\)
In the given figure, what is the measure of q?
- (a)
48°
- (b)
90°
- (c)
42°
- (d)
110°
Clearly p = 360° - 270° = 90° (Angles at a point)
Through C, draw a line l parallel to AB and DE.
\(\therefore\)42°+x = 180° and q +y = 180°
\(\Rightarrow\)x = 180° - 42° = 138°
\(\therefore\) y = 270° -138° = 132°
\(\therefore\) q = 180° -132° = 48°
In the given diagram (not drawn to scale), PO and RS are straight lines. Which of tile following statements is true?
- (a)
\(\angle f=\angle d\)
- (b)
\(\angle c=\angle e\)
- (c)
\(\angle a+\angle b=\angle f+\angle e\)
- (d)
\(\angle b+\angle c=\angle e+\angle f\)
\(\angle ROQ=\angle POS\Rightarrow \angle b+\angle c=\angle f+\angle e\)
(Vertically opposite angles)
In the figure (not drawn to scale), ABC and CDE are straight lines,\(\angle ACE\) is a right angle and DFIICGIIBH. Find \(\angle Y.\)
- (a)
65°
- (b)
73°
- (c)
62°
- (d)
60°
Since, CGIIDF and CE is transversal.
\(\therefore \angle GCE=\angle FDE\)
\(\Rightarrow \angle GCE=28^0\)
Now,\(\angle ACE=90^0\)
\(\Rightarrow \angle ACG+\angle GCE=90^0\)
\(\Rightarrow \angle ACG+28^0=90^0\)
\(\therefore \angle ACG=90^0-28^0=62^0\)
Also, BHIICG and CA is transversal.
\(\therefore \angle Y=ACG\)(Corresponding angles)
\(\Rightarrow \angle Y=62^0\)
In the figure (not drawn to scale), AJE, BJF, CJG and DJH are straight lines.
Find \(\angle x\ and \angle y\) respectively.
- (a)
50°, 49°
- (b)
59°, 40°
- (c)
59°,59°
- (d)
49°,48°
As, \(\angle AJB+\angle BJC+\angle CJD+\angle DJE=180^0\)(Angles on a straight line AE)
\(\therefore 42^0+\angle x+40^0+39^0=180^0\)
\(\Rightarrow \angle x=180^0-(42^0+40^0+39^0)\)
\(\Rightarrow \angle x=180^0-121^0=59^0\)
\(\angle y=\angle x\) (Vertically opposite angles)
\(\therefore \angle y=59^0\)
In the figure, PO is parallel to ST. AB is a straight line. Find \(\angle BST.\)
- (a)
110°
- (b)
125°
- (c)
152°
- (d)
98°
Since, PQ||ST
\(\therefore \angle PQS=\angle QST\) (Alternate angles)
\(\Rightarrow \angle QST=98^0\)
\(\Rightarrow QSA+\angle AST=98^0\)
\(\Rightarrow \angle AST=98^0-28^0=70^0\)
Now, AB is a straight line
\(\therefore \angle AST+\angle TSB=180^0\) (linear pair)
\(\Rightarrow \angle TSB=180^0-70^0=110^0\)
In the given figure (not drawn to scale), \(\angle UVT=72^0\ and\angle TSZ=53^0\) Find the value of \(\angle XZY+\angle SXY.\)
- (a)
60°
- (b)
125°
- (c)
180°
- (d)
None of these
We have,
XZ II UVand ST II XY.
\(\therefore \angle UVY=\angle XZY\) (Corresponding angles)
\(\Rightarrow \angle XZY=72^0\)
Also,\(\angle TSZ=\angle YXS\) (Corresponding angles)
\(\Rightarrow \angle YXS=53^0\)
Hence,\(\angle XZY+\angle YXS=72^0+53^0=125^0\)
In the given figure, AB II GHII DE, GF II BD II HI and \(\angle FCG=80^0.Find\ the\ value\ of \angle CHI.\)
- (a)
800
- (b)
120°
- (c)
1000
- (d)
1600
GF II HI and GH is the transversal line
So,\(\angle FGC=\angle CHI\) (Alternate interior angles)
\(\therefore \angle CHI=80^0\)
In the given figure, PQ, RS and UT are parallel lines. If c = 75° and a = (2/5)c, find b + d/2.
- (a)
92°
- (b)
115°
- (c)
112.5°
- (d)
135.5°
We have, C = 75°
\(\therefore a=\frac{2}{5}c=\frac{2}{5}\times75^0=30^0\)
Now, UT II PQ
\(\Rightarrow\) c = a + b (Alternate angles)
\(\Rightarrow\) 75° = 30° + b \(\Rightarrow\) b = 75° - 30° = 45°
Also, PQ II RS
\(\therefore\) b + d = 180° (Co-interior angles)
\(\Rightarrow\) d= 180°-45°= 135°
So,\(b+\frac{d}{2}=45^0+\frac{135^0}{2}=45^0+67.5^0=112.5^0\)
In the given figure, AB and CD are straight lines. Find \(\angle y.\)
- (a)
970
- (b)
270
- (c)
770
- (d)
550
Since, CD is a straight line.
\(\therefore 40^0+35^0\angle BOC=180^0\)
\(\Rightarrow \angle BOC=180^0-75^0=105^0\)
Now, AB is a straight line
\(\therefore \angle BOC+\angle y+48^0=180^0\)
\(\therefore \angle y=180^0-(105^0+48^0)=27^0\)
In the figure (not drawn to scale), DAE, CBH and ACG are straight lines, DAE II CBH II FG. Find x and y respectively.
- (a)
70°,35°
- (b)
110°,145°
- (c)
110°,35°
- (d)
140°,20°
Since, FG II DAE and AG is transversal.
\(\therefore \angle FGA+\angle DAG=180^0\)(Co-interior angles)
\(\Rightarrow 70^0+\angle x=180^0\Rightarrow \angle x=110^0\)
Also, CBHIIDAE and BA is transversal.
\(\therefore \angle y+35^0=180^0\) (Co-interior angles)
\(\Rightarrow \angle y=180^0-35^0\Rightarrow \angle y=145^0\)
Fill in the blanks.
(i) A _ P____ has two end points.
(ii) A line has ___Q______ end points on either side.
(iii) A___R____is a line that intersects two or more lines at distinct points.
(iv) An____S____is formed when two rays meet.
- (a)
P Q R S Line Two Ray Angle - (b)
P Q R S Line
segmentNo Transversal Angle - (c)
P Q R S Ray No Transversal Line - (d)
P Q R S Line
segmentTwo Transversal Angle
Mohit got an assignment where he had to explain the types of angles formed by a pair of parallel lines.
He completed the assignment but when his teacher checked he saw mistakes. Identify the correct option pointing out the mistake in the parts (i to iv).
Types of Angle | Observation |
(i) Corresponding | \(\angle 1=\angle 5,\angle 3=\angle 7,\angle 2=\angle 6,\angle 4=\angle 8\) |
(ii) Alternate interior | \(\angle 3=\angle 6,\angle 4=\angle 5\) |
(iii) Vertically opposite | \(\angle 1=\angle 4,\angle 5=\angle 6\) |
(iv) Alternate exterior | \(\angle 1=\angle 8,\angle 2=\angle 5\) |
- (a)
Only (i)
- (b)
Only (ii)
- (c)
Both (i) and (iii)
- (d)
Both (iii) and (iv)
(iii) \(\angle 1=\angle 4,\angle 5=\angle 8\) are vertically opposite angles.
Also,(iv) \(\angle 1=\angle 8\ and\ \angle 2=\angle 7\) are alternate exterior angles.