Euclidean Geometry | Jamb Mathematics

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Hello! As you prepare for your examination on Euclidean Geometry, focus on understanding fundamental concepts such as points, lines, angles, triangles, circles, and their properties. Be sure to practice theorem applications, problem-solving techniques, and geometric proofs to strengthen your grasp of the topic.

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Euclidean Geometry? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Euclidean Geometry together and move one step closer to achieving your exam success! Blissful learning.

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Calculation problem involving lines and angles

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1. Sum of Angles in a Triangle

Question: In a triangle, two angles measure

50^\circ

and

60^\circ

. Find the third angle.

Solution:
The sum of the angles in a triangle is

180^\circ

.
Let the third angle be

x

50^\circ + 60^\circ + x = 180^\circ

110^\circ + x = 180^\circ

x = 180^\circ - 110^\circ = 70^\circ

.
Thus, the third angle is $70^\circ$ .

2. Complementary Angles

Question: If one angle measures

35^\circ

, find its complement.

Solution:
Complementary angles sum to

90^\circ

.
Let the unknown angle be

x

35^\circ + x = 90^\circ

x = 90^\circ - 35^\circ = 55^\circ

.
Thus, the complement is $55^\circ$ .

3. Supplementary Angles

Question: If one angle is

110^\circ

, find its supplement.

Solution:
Supplementary angles sum to

180^\circ

.
Let the unknown angle be

x

110^\circ + x = 180^\circ

x = 180^\circ - 110^\circ = 70^\circ

.
Thus, the supplement is $70^\circ$ .

4. Adjacent Angles

Question: Two adjacent angles share a common side. If one angle is

65^\circ

and the other is

25^\circ

, find their sum.

Solution:
Sum of adjacent angles =

65^\circ + 25^\circ = 90^\circ

.
Thus, the total is $90^\circ$ .

5. Vertically Opposite Angles

Question: Two vertically opposite angles are equal. If one measures

72^\circ

, find the other.

Solution:
Vertically opposite angles are equal:
Thus, the second angle is also $72^\circ$ .

6. Parallel Lines and Transversals

Question: In a pair of parallel lines cut by a transversal, one alternate interior angle measures

40^\circ

. Find the other alternate interior angle.

Solution:
Alternate interior angles are equal.
Thus, the other angle is also $40^\circ$ .

7. Corresponding Angles

Question: In a set of parallel lines cut by a transversal, one corresponding angle measures

50^\circ

. Find its corresponding angle.

Solution:
Corresponding angles are equal.
Thus, the angle is also $50^\circ$ .

8. Perpendicular Lines

Question: If two lines are perpendicular, what is the measure of the angle they form?

Solution:
Perpendicular lines form a right angle, which is $90^\circ$ .

9. Interior Angles of a Quadrilateral

Question: The angles of a quadrilateral are

80^\circ

95^\circ

, and

100^\circ

. Find the fourth angle.

Solution:
Sum of angles in a quadrilateral =

360^\circ

.
Let the unknown angle be

x

80^\circ + 95^\circ + 100^\circ + x = 360^\circ

275^\circ + x = 360^\circ

x = 360^\circ - 275^\circ = 85^\circ

.
Thus, the missing angle is $85^\circ$ .

10. Angle Between Two Perpendicular Bisectors

Question: Two perpendicular bisectors intersect. What is the angle between them?

Solution:
Since they are perpendicular, they form a $90^\circ$ angle.

11. Sum of Exterior Angles of a Polygon

Question: What is the sum of the exterior angles of any polygon?

Solution:
The sum of the exterior angles of any polygon is always $360^\circ$ .

12. Right Triangle Complementary Angles

Question: One acute angle of a right triangle is

37^\circ

. Find the other acute angle.

Solution:
In a right triangle, the two acute angles sum to

90^\circ

.
Let the unknown angle be

x

37^\circ + x = 90^\circ

x = 90^\circ - 37^\circ = 53^\circ

.
Thus, the missing angle is $53^\circ$ .

13. Exterior Angle Theorem

Question: In a triangle, one exterior angle measures

110^\circ

. If one of the opposite interior angles is

40^\circ

, find the other interior angle.

Solution:
Exterior angle theorem states:

110^\circ = 40^\circ + x

x = 110^\circ - 40^\circ = 70^\circ

.
Thus, the missing interior angle is $70^\circ$ .

14. Reflex Angle

Question: Find the reflex angle of

120^\circ

Solution:
Reflex angle =

360^\circ - 120^\circ = 240^\circ

.
Thus, the reflex angle is $240^\circ$ .

15. Bisected Angle

Question: An angle of

80^\circ

is bisected. Find each resulting angle.

Solution:
Each angle =

\frac{80^\circ}{2} = 40^\circ

.
Thus, each angle is $40^\circ$ .

16. Parallel Line Alternate Exterior Angles

Question: If an alternate exterior angle measures

130^\circ

, find its pair.

Solution:
Alternate exterior angles are equal.
Thus, the other angle is also $130^\circ$ .

17. Linear Pair

Question: Two angles form a linear pair. If one angle is

112^\circ

, find the other.

Solution:
Linear pairs sum to

180^\circ

.
Let the unknown angle be

x

112^\circ + x = 180^\circ

x = 180^\circ - 112^\circ = 68^\circ

.
Thus, the other angle is $68^\circ$ .

18. Angle in an Equilateral Triangle

Question: Find each angle in an equilateral triangle.

Solution:
Each angle in an equilateral triangle is $60^\circ$ .

19. Opposite Angles in a Parallelogram

Question: If one angle in a parallelogram is

75^\circ

, find its opposite angle.

Solution:
Opposite angles in a parallelogram are equal.
Thus, the opposite angle is also $75^\circ$ .

20. Sum of Interior Angles of a Hexagon

Question: Find the sum of the interior angles of a hexagon.

Solution:
Sum of interior angles of an

n

-sided polygon =

(n - 2) \times 180^\circ

.
For a hexagon (

n = 6

(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.
Thus, the sum is $720^\circ$ .

Calculation problem invloving polygons

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1. Sum of Interior Angles of a Triangle

Question: Find the sum of the interior angles of a triangle.

Solution:
The formula for the sum of interior angles of a polygon is:

S = (n - 2) \times 180^\circ

, where

n

is the number of sides.
For a triangle (

n = 3

S = (3 - 2) \times 180^\circ = 180^\circ

.
Thus, the sum of the interior angles is $180^\circ$ .

2. Sum of Interior Angles of a Quadrilateral

Question: Find the sum of the interior angles of a quadrilateral.

Solution:

S = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ

.
Thus, the sum is $360^\circ$ .

3. Sum of Interior Angles of a Pentagon

Question: Find the sum of the interior angles of a pentagon.

Solution:

S = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.
Thus, the sum is $540^\circ$ .

4. Sum of Interior Angles of a Hexagon

Question: Find the sum of the interior angles of a hexagon.

Solution:

S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.
Thus, the sum is $720^\circ$ .

5. Sum of Interior Angles of an Octagon

Question: Find the sum of the interior angles of an octagon.

Solution:

S = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.
Thus, the sum is $1080^\circ$ .

6. Measure of One Interior Angle of a Regular Pentagon

Question: Find the measure of one interior angle of a regular pentagon.

Solution:
A regular polygon has all angles equal. The formula for one interior angle is:

\theta = \frac{S}{n} = \frac{540^\circ}{5} = 108^\circ

.
Thus, each interior angle is $108^\circ$ .

7. Measure of One Interior Angle of a Regular Hexagon

Question: Find the measure of one interior angle of a regular hexagon.

Solution:

\theta = \frac{720^\circ}{6} = 120^\circ

.
Thus, each interior angle is $120^\circ$ .

8. Measure of One Interior Angle of a Regular Octagon

Question: Find the measure of one interior angle of a regular octagon.

Solution:

\theta = \frac{1080^\circ}{8} = 135^\circ

.
Thus, each interior angle is $135^\circ$ .

9. Sum of Exterior Angles of Any Polygon

Question: What is the sum of the exterior angles of any polygon?

Solution:
For any polygon, the sum of exterior angles is always $360^\circ$ .

10. Measure of One Exterior Angle of a Regular Pentagon

Question: Find the measure of one exterior angle of a regular pentagon.

Solution:

\theta_{\text{ext}} = \frac{360^\circ}{5} = 72^\circ

.
Thus, each exterior angle is $72^\circ$ .

11. Measure of One Exterior Angle of a Regular Hexagon

Question: Find the measure of one exterior angle of a regular hexagon.

Solution:

\theta_{\text{ext}} = \frac{360^\circ}{6} = 60^\circ

.
Thus, each exterior angle is $60^\circ$ .

12. Measure of One Exterior Angle of a Regular Octagon

Question: Find the measure of one exterior angle of a regular octagon.

Solution:

\theta_{\text{ext}} = \frac{360^\circ}{8} = 45^\circ

.
Thus, each exterior angle is $45^\circ$ .

13. Number of Sides Given Interior Angle

Question: A regular polygon has each interior angle measuring

140^\circ

. Find the number of sides.

Solution:
Using the formula:

\theta_{\text{int}} + \theta_{\text{ext}} = 180^\circ

140^\circ + \theta_{\text{ext}} = 180^\circ

\theta_{\text{ext}} = 40^\circ

Now, using

\theta_{\text{ext}} = \frac{360^\circ}{n}

40^\circ = \frac{360^\circ}{n}

n = \frac{360}{40} = 9

.
Thus, the polygon has 9 sides.

14. Number of Diagonals in a Pentagon

Question: Find the number of diagonals in a pentagon.

Solution:
The formula for the number of diagonals in a polygon is:

D = \frac{n(n - 3)}{2}

.
For a pentagon (

n = 5

D = \frac{5(5 - 3)}{2} = \frac{5(2)}{2} = 5

.
Thus, a pentagon has 5 diagonals.

15. Number of Diagonals in a Hexagon

Question: Find the number of diagonals in a hexagon.

Solution:

D = \frac{6(6 - 3)}{2} = \frac{6(3)}{2} = 9

.
Thus, a hexagon has 9 diagonals.

16. Number of Diagonals in an Octagon

Question: Find the number of diagonals in an octagon.

Solution:

D = \frac{8(8 - 3)}{2} = \frac{8(5)}{2} = 20

.
Thus, an octagon has 20 diagonals.

17. Find a Missing Interior Angle in a Quadrilateral

Question: A quadrilateral has three angles measuring

90^\circ

85^\circ

, and

75^\circ

. Find the fourth angle.

Solution:
Sum of interior angles =

360^\circ

90^\circ + 85^\circ + 75^\circ + x = 360^\circ

x = 360^\circ - 250^\circ = 110^\circ

.
Thus, the missing angle is $110^\circ$ .

18. Find a Missing Interior Angle in a Pentagon

Question: A pentagon has four angles measuring

100^\circ

120^\circ

95^\circ

, and

85^\circ

. Find the fifth angle.

Solution:
Sum of interior angles =

540^\circ

100^\circ + 120^\circ + 95^\circ + 85^\circ + x = 540^\circ

x = 540^\circ - 400^\circ = 140^\circ

.
Thus, the missing angle is $140^\circ$ .

19. Relationship Between Interior and Exterior Angles

Question: Find the relationship between an interior and exterior angle in any polygon.

Solution:
Each interior and exterior angle are supplementary:

\theta_{\text{int}} + \theta_{\text{ext}} = 180^\circ

20. Regular vs Irregular Polygons

Question: How do you differentiate a regular polygon from an irregular polygon?

Solution:
A regular polygon has equal sides and angles, whereas an irregular polygon has unequal sides or angles.

Calculation problems involving angles using circle theorems

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Here are 20 unique calculation problems involving angles using Circle Theorems, with step-by-step solutions following your LaTeX formatting instructions.

1. Angle in a Semicircle

Question: A triangle is inscribed in a semicircle where the diameter is the base. If one angle of the triangle is

x^\circ

, find

x

Solution:
The angle in a semicircle theorem states that an angle subtended by a semicircle is always $90^\circ$ .
Thus,

x = 90^\circ

2. Angles in the Same Segment

Question: In a circle, two angles subtended by the same chord on the same segment are

45^\circ

and

y^\circ

. Find

y

Solution:
Angles in the same segment are equal.
Thus,

y = 45^\circ

3. Angle at the Center vs Angle at the Circumference

Question: An angle subtended at the center of a circle is

120^\circ

. Find the angle subtended by the same arc at the circumference.

Solution:
The angle at the center is twice the angle at the circumference.
Let the angle at the circumference be

x

x = \frac{120^\circ}{2} = 60^\circ

.
Thus, the angle is $60^\circ$ .

4. Alternate Segment Theorem

Question: A tangent to a circle makes an angle of

40^\circ

with a chord. Find the angle in the alternate segment.

Solution:
By the alternate segment theorem, the angle in the alternate segment is equal to the angle between the tangent and the chord.
Thus, the angle is $40^\circ$ .

5. Opposite Angles in a Cyclic Quadrilateral

Question: A cyclic quadrilateral has one angle measuring

110^\circ

. Find its opposite angle.

Solution:
The sum of opposite angles in a cyclic quadrilateral is $180^\circ$ .
Let the unknown angle be

x

110^\circ + x = 180^\circ

x = 180^\circ - 110^\circ = 70^\circ

.
Thus, the opposite angle is $70^\circ$ .

6. Cyclic Quadrilateral Angle

Question: A cyclic quadrilateral has angles measuring

80^\circ

x^\circ

95^\circ

, and

y^\circ

. Find

x

and

y

Solution:
Opposite angles in a cyclic quadrilateral sum to $180^\circ$ .

For

x

x + 80^\circ = 180^\circ

x = 100^\circ

For

y

y + 95^\circ = 180^\circ

y = 85^\circ

Thus,

x = 100^\circ

and

y = 85^\circ

7. Tangent and Radius

Question: A radius of a circle meets a tangent at a point. What is the angle between them?

Solution:
The tangent and radius theorem states that the radius meets the tangent at $90^\circ$ .

8. Perpendicular from the Center

Question: A perpendicular is drawn from the center of a circle to a chord. How does it affect the chord?

Solution:
The perpendicular from the center to a chord bisects the chord.

9. Angle Between Two Tangents

Question: Two tangents are drawn to a circle from an external point. If the angle between them is

50^\circ

, find the angle subtended at the center.

Solution:
The external angle is half the angle subtended at the center.
Let the angle at the center be

x

x = 2 \times 50^\circ = 100^\circ

.
Thus, the angle is $100^\circ$ .

10. Chord and Perpendicular Bisector

Question: A chord is bisected by a perpendicular from the center. If one half of the chord is

5

cm and the perpendicular is

12

cm, find the radius of the circle.

Solution:
Using Pythagoras' Theorem:

r^2 = 12^2 + 5^2

r^2 = 144 + 25

r = \sqrt{169} = 13

cm.
Thus, the radius is $13$ cm.

11. Length of a Tangent

Question: A circle has a radius of

7

cm, and an external point is

25

cm from the center. Find the length of the tangent.

Solution:
Using Pythagoras' theorem:

t^2 + 7^2 = 25^2

t^2 + 49 = 625

t^2 = 576

t = \sqrt{576} = 24

cm.
Thus, the tangent length is $24$ cm.

12. Two Tangents from a Point

Question: Two tangents from an external point to a circle are equal. One is

10

cm long. Find the other.

Solution:
By the two tangents theorem, both tangents from an external point are equal.
Thus, the second tangent is $10$ cm.

13. Exterior Angle of a Cyclic Quadrilateral

Question: An exterior angle of a cyclic quadrilateral is

75^\circ

. Find the interior opposite angle.

Solution:
An exterior angle is equal to the opposite interior angle in a cyclic quadrilateral.
Thus, the opposite angle is $75^\circ$ .

14. Angle in the Major Segment

Question: An angle in a major segment is given as

120^\circ

. Find the angle in the minor segment.

Solution:
Angles in the same segment add up to $180^\circ$ .
Thus, the minor segment angle is $60^\circ$ .

15. Interior Angle Between Two Chords

Question: Two chords intersect inside a circle at an angle of

130^\circ

. Find the two exterior angles.

Solution:
The exterior angles are supplementary to the interior angle.

180^\circ - 130^\circ = 50^\circ

.
Thus, each exterior angle is $50^\circ$ .

16. Angle in an Inscribed Equilateral Triangle

Question: A circle has an equilateral triangle inscribed in it. Find each angle.

Solution:
Each angle in an equilateral triangle is $60^\circ$ .

17. Two Chords Intersecting Inside a Circle

Question: Two chords intersect at a point inside a circle forming angles of

40^\circ

and

x^\circ

. Find

x

Solution:
Opposite angles are equal, so

x = 40^\circ

18. Reflex Angle at the Center

Question: If an angle subtended at the center is

100^\circ

, find its reflex angle.

Solution:
Reflex angle =

360^\circ - 100^\circ = 260^\circ

19. Alternate Segment Angle in a Tangent-Chord

Question: A tangent to a circle forms a

58^\circ

angle with a chord. Find the angle in the alternate segment.

Solution:
By the alternate segment theorem, the angle in the alternate segment is also $58^\circ$ .

20. Finding the Angle Between Two Radii

Question: Two radii form an isosceles triangle in a circle. If the base angle is

50^\circ

, find the angle at the center.

Solution:
Using the sum of angles in a triangle:

x + 50^\circ + 50^\circ = 180^\circ

x = 180^\circ - 100^\circ = 80^\circ

.
Thus, the angle at the center is $80^\circ$ .

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Euclidean Geometry | Jamb Mathematics

Calculation problem involving lines and angles

1. Sum of Angles in a Triangle

2. Complementary Angles

3. Supplementary Angles

4. Adjacent Angles

5. Vertically Opposite Angles

6. Parallel Lines and Transversals

7. Corresponding Angles

8. Perpendicular Lines

9. Interior Angles of a Quadrilateral

10. Angle Between Two Perpendicular Bisectors

11. Sum of Exterior Angles of a Polygon

12. Right Triangle Complementary Angles

13. Exterior Angle Theorem

14. Reflex Angle

15. Bisected Angle

16. Parallel Line Alternate Exterior Angles

17. Linear Pair

18. Angle in an Equilateral Triangle

19. Opposite Angles in a Parallelogram

20. Sum of Interior Angles of a Hexagon

Calculation problem invloving polygons

1. Sum of Interior Angles of a Triangle

2. Sum of Interior Angles of a Quadrilateral

3. Sum of Interior Angles of a Pentagon

4. Sum of Interior Angles of a Hexagon

5. Sum of Interior Angles of an Octagon

6. Measure of One Interior Angle of a Regular Pentagon

7. Measure of One Interior Angle of a Regular Hexagon

8. Measure of One Interior Angle of a Regular Octagon

9. Sum of Exterior Angles of Any Polygon

10. Measure of One Exterior Angle of a Regular Pentagon

11. Measure of One Exterior Angle of a Regular Hexagon

12. Measure of One Exterior Angle of a Regular Octagon

13. Number of Sides Given Interior Angle

14. Number of Diagonals in a Pentagon

15. Number of Diagonals in a Hexagon

16. Number of Diagonals in an Octagon

17. Find a Missing Interior Angle in a Quadrilateral

18. Find a Missing Interior Angle in a Pentagon

19. Relationship Between Interior and Exterior Angles

20. Regular vs Irregular Polygons

Calculation problems involving angles using circle theorems

1. Angle in a Semicircle

2. Angles in the Same Segment

3. Angle at the Center vs Angle at the Circumference

4. Alternate Segment Theorem

5. Opposite Angles in a Cyclic Quadrilateral

6. Cyclic Quadrilateral Angle

7. Tangent and Radius

8. Perpendicular from the Center

9. Angle Between Two Tangents

10. Chord and Perpendicular Bisector

11. Length of a Tangent

12. Two Tangents from a Point

13. Exterior Angle of a Cyclic Quadrilateral

14. Angle in the Major Segment

15. Interior Angle Between Two Chords

16. Angle in an Inscribed Equilateral Triangle

17. Two Chords Intersecting Inside a Circle

18. Reflex Angle at the Center

19. Alternate Segment Angle in a Tangent-Chord

20. Finding the Angle Between Two Radii

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