Geometry

Consider the points named A, B, C, D, and E. such that no three of them are collinear. It can simply be assumed to form a pentagon. Now draw the diagonals in the figure and count the total number of lines including sides of pentagon.

So the lines are AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.

So that makes 10 lines.

Now, let’s start counting lines.

AB, AC, AD, AE

BC, BE (BD would be same as BC and BA is covered in above in AB)

CE

DE

∴ there are 8 combinations

There is a formula to find the number of diagonals in any n sided polygon.

where d is the number of diagonals.

This formula could be easily proved by basic knowledge of permutations and combinations.

If you don’t know about it, its fine you’ll learn it in higher classes for now you may just memorise this small formula or you can also count the number of diagonals.

So here if we put n = 7 ∵ heptagon has 7 sides

we get d = 14

we can simply count it

remember AB is same as BA.

So the line segments are

AB AC AD AE

BC BD BE

CD CE

DE

A complete circle is of 360° and a clock completes 12 hours in 1 cycle.

When its 9 O’ clock then

hour hand is at 9 and minute hand is at 12.

So, let’s consider the angle by minute hand to be 0 i.e., assuming 12 as the beginning point of counting.

∴ is the angle covered by hour hand.

⇒ Angle between hour hand and minute hand is 270 – 0 = 270°

is the other direction it is 360-270 = 90°.

⇒ B is correct.

The total angle is 360° as it is a complete circle.

Now there are 48 spokes which also means

the angle between two consecutive spokes is

⇒ C is correct

As in ∠ZXY X is in the middle and it signifies representation of ∠X enclosed by Z and Y.

∵ the points B and P and Y are not moving

the ∠BPY will always be same.

There are 3 individual angles

Then 2 angles in pair of 2, i.e., 40+20 and 20+30 (40+30 is not considered as it is not continuous)

And then there is one angle which is the pair of all three i.e., 40+20+30.

So in total there are 6 different angles.

Obtuse angles are the angles which are greater than 90° and less than 180°.

so here individually none of them is an obtuse angle.

if we combine 20° and 45° still it is not and obtuse angle.

but, if combine 20° + 45° + 65°

then it is an obtuse angle.

also 20° + 45° + 65° + 30° is not an obtuse angle since its equal to 180°.

Similarly, 65 + 30 and 45 + 65 + 30 are also obtuse.

and 45° + 65° is also obtuse.

∴ 4 angles are obtuse

1 is the bigger equilateral triangle, then there are 4 other smaller equilateral triangle and then the altitude makes 2 new big right angle triangles and 2 smaller and 1 large triangle on each side.

total 13 triangles

D. is not possible as sum of right angles is always equal to 180°.

D is not possible as sum of right angles is always equal to 180° but to be obtuse it should be between 90° and less than 180°.

2 least consecutive prime numbers are 2 and 3.

⇒ No. of sides = 5

⇒

⇒ d = 5

B.5 is correct

ABC is isosceles so are ABD and CBD.

⇒ 3 triangles are isosceles

BAC, ADC, ADB

B as its two sides are equal and one angle is 90°

reflex angle

reflex angle, angles greater than 180° are reflex angles.

9, use the formula

parallel

trapezium have two parallel and non-parallel sides.

O and S,

N and T,

M, P, Q and R

We can see in the diagram the point inside the outlines and points outside the outline.

(a) BD, (b) CD, (c) C, (d) C, (e) 5

Check for the part which is remaining.

L.H.S = R.H.S

(a) Right

(b) acute

(c) obtuse

∠AOD = 90 (40+20+30)

∠COA = 50 (20+30) (less than 90, which makes it acute)

∠AOE = 130 (40+40+20+30) (between 90 and 180 which makes it obtuse.)

5

ΔAOB, ΔCOD, ΔAOC, ΔACD, ΔCAB.

12

∠OAB,∠OBA,∠AOB,∠AOC,∠OAC,∠OCA,∠OCD,∠COD,∠ODC,∠BAC,∠DCA,∠DOB.

4

∠AOD, ∠BOC, ∠DOA, ∠COB

2, 4

Straight line have 180°, complete angle have 360° and Right angle have 90°.

⇒ 2×90 = 180

⇒ 4×90 = 360

2

Only P and Q are common

1

Only point A is common.

3

They are P, Q and R.

4

They are D, E, F, and G.

AB

Check the figure in BAC and BAD, BA is common which is same as AB.

True

Lines that never slant up or down are called horizontal lines. Lines that go straight up and down are called vertical lines. Therefore, when ever they intersect, they intersect at right angles.

False

If the size of the arms changes, then there will be no change in the measure of the angle formed by those arms.

False

If the size of the arms changes, then there will be no change in the measure of the angle formed by those arms.

True

If a line is parallel to another line, then their parts are also parallel.

False

By definition, parallel lines are those which never intersect each other.

True

Because in both measurements ∠ ABC and ∠ CBA, the common angle is B.

∴ ∠ ABC = ∠ CBA

False

Two line segments will intersect each other at only one point.

False

Only one line can pass through two given point.

False

Many lines can pass through a given point.

False

Two angles can have either one or two points in common.

A line segment is a part of line having finite length.

Hence, all the line segment shown in the figure are AB, AC, AD, AE, BC, BD, BE, CD, CE and DE.

There are five line segments in the given figure, namely AB, BC, CD, DE and AE.

Mid – point of a line segment divides it into two equal parts.

Clearly, from the figure,

AZ = ZB, AX = XC and CY = YB. So, Z, X and Y are the mid-points of AS, AC and CB respectively.

Hence, there are 3 mid-points, i.e. X, Z and Y.

There are five vertices in the given figure, namely A, B, C, D and E and there are seven line segments in given Figure, namely AB, BC, CD, DE, AE, AD and AC.

The fifteen angles (less than 180°) involved in the figure:

∠ EAD, ∠ AEF, ∠ EFD, ∠ ADF, ∠ DFC, ∠ DCF, ∠ CDF, ∠ BEF, ∠ BFE, ∠ EBF, ∠ FBC, ∠ FCB, ∠ BFC, ∠ ABC, and ∠ ACB.

(a) ∠ 1 = ∠ CBD

(b) ∠ 2= ∠ DBE

(c) ∠ 3 = ∠ EBA

(d) ∠ 1 +∠ 2 = ∠ CBD + ∠ DBE

= ∠ CBE

(e) ∠ 2 + ∠ 3 = ∠ DBE + ∠ EBA

= ∠ DBA

(f) ∠ 1 +∠ 2 + ∠ 3 = ∠ CBD + ∠ DBE + ∠ EBA

= ∠ CBA or ∠ ABC

(g) ∠ CBA - ∠ 1 = ∠ CBA - ∠ CBD

= ∠ DBA or ∠ ABD

(i). Points A, B and C

Line segment AB, BC and AC

(ii). Points A, B, C and D

Line segment AB, BC, CD and AD

(iii). Points A, B, C, D and E

Line segment AB, BC, CD, DE and AE

(iv). Points A, B, C, D, E and F

Line segment AB, CD, and EF

In figure (ii), point O appears to be the mid-point and equal of the line segments formed OA and OB.

Also, in figure (iii), Point D appears to be the mid-point and equal line segments formed are BD and DC.

(a) No, it is not possible that the same line segments have two different lengths

(b) No, it is not possible that the same angles have different measure.

Yes, because ∠ ABC and ∠ CBD together form ∠ ABD, i.e. ∠ ABC + ∠ CBD = ∠ ABD.

Yes, because the line segments AB and BC together form the line segment AC. i.e. AB + BC = AC

Angles are measured in degrees. The symbol for degree is a little circle.

The full circle is 360° (360 degree). A half circle or a straight angle is 180°. A quarter circle or a right angle is 90°.

Place the mid-point of the protractor on the Vertex of the angle. Line up one side of the angle with the Zero line of the protractor (where you see the number 0).

Read the degrees here the other side crosses the number scale.

1. Measure the angles.

2. Measure the angles. Label each angle as acute or obtuse.

3. Tasha measured an acute angle, and got 146°. The teacher pointed out that she had read the wrong set of numbers on the protractor.

4. Measure the following angles using your own protractor. If you need to, make the sides of the angles longer with a ruler.

5. Draw four dots and connect them so that you get a quadrilateral.

Measure all the angles of your quadrilateral. Then add the angle measure.

Yes, points 6 and C lie in the interior of ∠ 2 also. Since, ∠ 1 is in interior of ∠ 2, then all the points lying inside the ∠ 1, will also lie inside the ∠ 2.

Angle is made by two rays or lines having a common end point. So, option (b) and (c) are incorrect.

A bisector is a line which bisects a given line segment into two equal parts. If this bisector is perpendicular to the given line segment, then it is known as perpendicular bisector.

(a) Figure (ii) represents a perpendicular bisector.

(b) Figures (ii) and (iii) represents bisectors.

(c) Figure (iii) represents only bisector.

(d) Figure (i) represents only perpendicular.

Both the figures have three lines segments.

Figure (i) is not a triangle because it is not a closed figure.

No, because angle is made when two rays intersect at common point. The common point is known as vertex of an angle.

(a) The four angles that appear to be acute angles are ∠ AEB, ∠ ADE, ∠ BAE and ∠ BCE.

(b) ∠ BCD and ∠ BAD are angles that appear to be obtuse angles.

(a) Yes

(b) No, it is not possible

(c) No, it is not possible

(a) AE + EC = AC

(b) AC – EC = AE

(c) BD – BE = ED

(d) BD – DE = BE

A right angle is an angle of measure 90°. It is formed by two perpendicular lines.

(a) ∠ABD because BA ⊥ BD

(b) ∠ RTS because RT ⊥ ST

(c) ∠ ACD and ∠ ACB because AC ⊥ BD

(d) ∠ RTW and ∠ RTS because RT ⊥ SW

(e) ∠ AED, ∠ AEB, ∠ BEC and ∠ DEC because diagonals of rhombus are perpendicular to each other.
So AC ⊥ BD and E is their point of intersection.

(f) ∠ AEC because AE ⊥ CE as diagonals of rhombus are perpendicular to each other.

(g) ∠ ACD because AC ⊥ CD

(h) ∠ AKO, ∠ AKP, ∠ BKO and ∠ BKP because OP ⊥ AB and K is their point of intersection.

a) Since, DB is the bisector of ∠ ADC.

This means that DB divides ∠ ADC into 2 equal parts.

∴ ∠ ADB = ∠ BDC

b) Since, BD bisects ∠ ABC.

This means that, ∠ ABD = ∠ DBC.

c) Since, DC is the bisector of ∠ ADB

∴ ∠ ADC = ∠ CDB ………….. (1)

Also, it is given that,

∠ CAD = ∠ CBD = 90° ……….. (2)

Also, we know that sum of interior angles of a triangle is equal to 180°.

∴ In Δ ACD,

∠ ACD+∠ CDA+ ∠ DAC =180° …. (3)

In Δ BCD,

∠ BCD+∠ CDB+∠ DBC =180° …. (4)

From (1), (2), (3), (4) we get,

∠ ACD = ∠ BCD

Trisectors are the lines which trisect an angle (or which divide an angle into 3 equal parts).

So, here we have two trisectors for ∠ BAE: AC and AD.

There are 2 points: A and B.

There is only one line segment: AB.

There are 3 points: A, B, C.

There are 3 line segments: AB, BC, and AC.

There are 4 points: A, B, C, D.

There are total 6 line segments:

AB, AC, AD, BC, BD, CD.

There are 5 points: A, B, C, D and E.

There are total 10 line segments:

AB, AD, AE, AC, BD, BE, BC, DE, DC, EC.

(a) Chords: PC and AB (longest chord i.e. diameter).

(b) Radii: OA, OB, OC and OP (radii are those whose one end point lies on the circumference of the circle and other end coincides with the centre of the circle.

(c) PC

(d)

Region shaded in green colour represents sector (OPB) and region shaded in red colour represents sector (OAC).

(e)

Region shaded in green represents the smaller segment of the circle formed by CP.

(a) Yes, we can have two acute angles whose sum is an acute angle.

We know that,

Acute angle is one that measures between 0° and 90°.

So, suppose we have two acute angles 30° and 40° then, sum of these two would be 70° and this lies between 0° and 90° therefore 70° is also acute .

You can think of more such examples.

[Note: basically, when two consider two acute angles smaller than 45° then their sum would always be smaller than 90° and so angle obtained will be acute.]

(b) Yes, we can have two acute angles whose sum is a right angle.

Example: 30° + 60° = 90°

45° + 45° = 90°

(c) Yes, we can have two acute angles whose sum is an obtuse angle.

We know that,

Acute angle is one that measures between 0° and 90°.

And obtuse angle is one that measures between 90° and 180°.

So, suppose we have two acute angles 50° and 60° then, sum of these two would be 110° and this lies between 90° and 180° therefore 110° is an obtuse angle .

You can think of more such examples.

[Note: basically, when two consider two acute angles greater than 45° then their sum would always be greater than 90° and so angle obtained will be obtuse.]

(d) No, we can have two acute angles whose sum is a straight angle.

We know that,

Acute angle is one that measures between 0° and 90°.

And straight angle measures 180°

If we take two 90° angles then we can obtain a straight angle but 90° is not an acute angle. So, in any case we cannot obtain a straight angle using two acute angles.

[Note: measure of acute angle is always less than 90° and on adding two 90° angles we are obtaining a straight angle. So, if we add any other acute angles then their sum would always be less than 180°.]

(e) No, we can have two acute angles whose sum is a straight angle.

We know that,

Acute angle is one that measures between 0° and 90°.

And reflex angle measures between 180° and 360°

If we take two 90° angles then we can obtain a 180° angle but 90° is not an acute angle. So, in any case we cannot obtain an angle which is greater than 180° (or a reflex angle).

[Note: measure of acute angle is always less than 90° and on adding two 90° angles we are obtaining a straight angle. So, if we add any other acute angles then their sum would always be less than 180°.]

a) Yes, we can have two obtuse angles whose sum is a reflex angle.

We know that,

Obtuse angle is one that measures between 90° and 180°.

Reflex angle is one that measures between 180° and 360°.

So, suppose we have two obtuse angles of measure 100° and 105° then their sum 205° lies between 180° and 360° and ∴ is a reflex angle.

You can think of more such examples.

b) No, we cannot have two obtuse angles whose sum is a complete angle.

We know that,

Complete angle measures 360° and obtuse angle is one that measures between 90° and 180°.

If we take two 180° angles then we can obtain a complete angle but 180° is not an obtuse angle. So, in any case we cannot obtain a complete angle using two obtuse angles.

a) Vertices: A, B, C, D, E, F

b) Edges: AB, BC, AC, DE, EF, DF, AE, BD, CF

c) Faces: Two triangular faces – ABC, DEF and three rectangular faces – BDFC, ACFE, ABDE

A sphere has a curved surface so, there are no flat surfaces.

∴ No. of Faces = 0

Also, there are no line segments or joints.

∴ Number of Edges = 0

And Number of Vertices = 0

So, Diagonals of the pentagon ABCDE are: AC, AD, BD, BE, CE (Shown by black line segments)