SSC GD Geometry

• Introduction to SSC GD Geometry pdf,
• 1. Lines and Angles
  • 1.1. What is a Line and a Ray?
  • 1.2. Types of Angles
• 2. Triangles (Basic Properties)
  • 2.1. Understanding Triangles
  • 2.2. Sum of Angles in a Triangle
• 3. Quadrilaterals (Basic Shapes)

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Introduction to SSC GD Geometry pdf,

Hello! We are starting our journey into Geometry for the SSC GD Constable 2026 exam. Geometry is about shapes, lines, and angles. It helps us understand the world around us, like how buildings are built or how a football field is shaped.

These notes are designed to be super simple, just like a SSC GD Geometry pdf, guide you can read quickly on your phone. We will learn the basic rules needed to solve the exam questions easily. Remember, practice makes perfect!

1. Lines and Angles

1.1. What is a Line and a Ray?

Geometry starts with a point. A point is just a dot, showing a location.

• Line: Imagine drawing a straight path that goes on forever in both directions. It never stops!
• Line Segment: This is just a small piece of a line. It has a clear start point and a clear end point. Think of a ruler.
• Ray: A ray has a starting point, but it goes on forever in only one direction. Think of a flashlight beam.

When two lines or rays meet at a point, they form an angle. Understanding these basics is key for solving problems in your SSC GD Geometry pdf, study material.

4 Solved Examples

Example 1: Two angles, $\mathrm{\angle }A$ and $\mathrm{\angle }B$, are on a straight line. If $\mathrm{\angle }A$ is ${70}^{\circ }$, what is the measure of $\mathrm{\angle }B$?

Solution:

1. First, let’s understand the rule: Angles on a straight line always add up to ${180}^{\circ }$. This is called a Linear Pair.
2. Now, write down the formula: $\mathrm{\angle }A+\mathrm{\angle }B={180}^{\circ }$
3. Put the numbers in: We know $\mathrm{\angle }A$ is ${70}^{\circ }$. So, ${70}^{\circ }+\mathrm{\angle }B={180}^{\circ }$
4. Calculate it: To find $\mathrm{\angle }B$, we subtract ${70}^{\circ }$ from ${180}^{\circ }$. $$\mathrm{\angle }B={180}^{\circ }-{70}^{\circ }$$ $$\mathrm{\angle }B={110}^{\circ }$$
5. So, the answer is: $\mathrm{\angle }B$ is ${110}^{\circ }$.

Example 2: Three angles meet at a point on a straight line. They are ${40}^{\circ }$, $x$, and ${60}^{\circ }$. Find the value of $x$.

Solution:

1. First, let’s understand what we need to find: We need to find the missing angle $x$. Since all three angles are on a straight line, their total must be ${180}^{\circ }$.
2. Now, write down the formula: ${40}^{\circ }+x+{60}^{\circ }={180}^{\circ }$
3. Combine the known numbers: ${40}^{\circ }+{60}^{\circ }={100}^{\circ }$. $${100}^{\circ }+x={180}^{\circ }$$
4. Calculate it: Subtract ${100}^{\circ }$ from ${180}^{\circ }$. $$x={180}^{\circ }-{100}^{\circ }$$ $$x={80}^{\circ }$$
5. So, the answer is: The missing angle $x$ is ${80}^{\circ }$.

Example 3: Two lines cross each other. One angle formed is ${55}^{\circ }$. What is the angle directly opposite to it?

Solution:

1. First, let’s understand the rule: When lines cross, the angles opposite each other are called Vertically Opposite Angles. They are always equal.
2. Now, write down the formula: Opposite Angle = Given Angle.
3. Put the numbers in: Given Angle is ${55}^{\circ }$.
4. Calculate it: The angle directly opposite must also be ${55}^{\circ }$.
5. So, the answer is: The opposite angle is ${55}^{\circ }$.

Example 4: A line segment is $15\text{ cm}$ long. It is divided into two parts. If the first part is $8\text{ cm}$, how long is the second part?

Solution:

1. First, let’s understand what we need to find: We need the length of the remaining piece.
2. Now, write down the formula: Total Length = Part 1 + Part 2. $$15\text{ cm}=8\text{ cm}+\text{Part 2}$$
3. Put the numbers in: $15=8+P_2$
4. Calculate it: Subtract the known part from the total length. $$P_2=15-8$$ $$P_2=7\text{ cm}$$
5. So, the answer is: The second part is $7\text{ cm}$ long. This simple logic applies to many problems in the SSC GD Geometry pdf,.

1.2. Types of Angles

Angles are measured in degrees (${}^{\circ }$). The size of the angle tells us its type.

• Acute Angle: Small angles, less than ${90}^{\circ }$. (Think of the sharp corner of a slice of pizza).
• Right Angle: Exactly ${90}^{\circ }$. (Think of the corner of a square table).
• Obtuse Angle: Bigger than ${90}^{\circ }$ but less than ${180}^{\circ }$. (Think of an open book).
• Straight Angle: Exactly ${180}^{\circ }$. (A straight line).

We also have special pairs of angles:

• Complementary Angles: Two angles that add up to exactly ${90}^{\circ }$.
• Supplementary Angles: Two angles that add up to exactly ${180}^{\circ }$. This is important for your SSC GD Geometry pdf, preparation.

4 Solved Examples

Example 1: Find the complement of an angle that measures ${35}^{\circ }$.

Solution:

1. First, let’s understand the rule: Complementary angles must add up to ${90}^{\circ }$.
2. Now, write down the formula: Complement $={90}^{\circ }-\text{Given Angle}$.
3. Put the numbers in: Complement $={90}^{\circ }-{35}^{\circ }$.
4. Calculate it: $$90-35={55}^{\circ }$$
5. So, the answer is: The complement is ${55}^{\circ }$.

Example 2: Find the supplement of an angle that measures ${125}^{\circ }$.

Solution:

1. First, let’s understand the rule: Supplementary angles must add up to ${180}^{\circ }$.
2. Now, write down the formula: Supplement $={180}^{\circ }-\text{Given Angle}$.
3. Put the numbers in: Supplement $={180}^{\circ }-{125}^{\circ }$.
4. Calculate it: $$180-125={55}^{\circ }$$
5. So, the answer is: The supplement is ${55}^{\circ }$.

Example 3: Two complementary angles are in the ratio $2:3$. Find the measure of the smaller angle.

Solution:

1. First, let’s understand what we need to find: The angles add up to ${90}^{\circ }$. We can call the angles $2x$ and $3x$.
2. Now, write down the formula: $2x+3x={90}^{\circ }$
3. Combine the terms: $5x={90}^{\circ }$
4. Calculate $x$: Divide ${90}^{\circ }$ by 5. $$x=90/5$$ $$x={18}^{\circ }$$
5. Find the smaller angle: The smaller angle is $2x$. $$2\times {18}^{\circ }={36}^{\circ }$$
6. So, the answer is: The smaller angle is ${36}^{\circ }$. This type of ratio problem is common in SSC GD Geometry pdf, exams.

Example 4: An angle is equal to its own complement. What is the measure of the angle?

Solution:

1. First, let’s understand what we need to find: Let the angle be $x$. Its complement is also $x$. They must add up to ${90}^{\circ }$.
2. Now, write down the formula: $x+x={90}^{\circ }$
3. Combine the terms: $2x={90}^{\circ }$
4. Calculate $x$: Divide ${90}^{\circ }$ by 2. $$x=90/2$$ $$x={45}^{\circ }$$
5. So, the answer is: The angle is ${45}^{\circ }$.

2. Triangles (Basic Properties)

2.1. Understanding Triangles

A triangle is a shape with three sides and three corners (vertices). Triangles are the strongest shapes in geometry.

• Sides: We name the sides $a,b,c$.
• Angles: We name the angles $\mathrm{\angle }A,\mathrm{\angle }B,\mathrm{\angle }C$.

There are different types of triangles based on their sides:

1. Equilateral: All 3 sides are equal, and all 3 angles are ${60}^{\circ }$.
2. Isosceles: Only 2 sides are equal, and the angles opposite those sides are equal.
3. Scalene: No sides are equal, and no angles are equal.

Mastering these properties is essential for your SSC GD Geometry pdf, success.

2.2. Sum of Angles in a Triangle

This is the most important rule for triangles:

The sum of all three interior angles in any triangle is always ${180}^{\circ }$.

$$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }$$

4 Solved Examples

Example 1: In a triangle, two angles are ${50}^{\circ }$ and ${65}^{\circ }$. Find the measure of the third angle ($x$).

Solution:

1. First, let’s understand the rule: All three angles must add up to ${180}^{\circ }$.
2. Now, write down the formula: ${50}^{\circ }+{65}^{\circ }+x={180}^{\circ }$
3. Combine the known numbers: ${50}^{\circ }+{65}^{\circ }={115}^{\circ }$. $${115}^{\circ }+x={180}^{\circ }$$
4. Calculate $x$: Subtract ${115}^{\circ }$ from ${180}^{\circ }$. $$x={180}^{\circ }-{115}^{\circ }$$ $$x={65}^{\circ }$$
5. So, the answer is: The third angle is ${65}^{\circ }$. (This is an Isosceles triangle!)

Example 2: The angles of a triangle are in the ratio $1:2:3$. Find the measure of the largest angle.

Solution:

1. First, let’s understand what we need to find: We can call the angles $1x$, $2x$, and $3x$. Their total is ${180}^{\circ }$.
2. Now, write down the formula: $1x+2x+3x={180}^{\circ }$
3. Combine the terms: $6x={180}^{\circ }$
4. Calculate $x$: Divide ${180}^{\circ }$ by 6. $$x=180/6$$ $$x={30}^{\circ }$$
5. Find the largest angle: The largest angle is $3x$. $$3\times {30}^{\circ }={90}^{\circ }$$
6. So, the answer is: The largest angle is ${90}^{\circ }$. This triangle is a Right-Angled triangle. This is a common question type in the SSC GD Geometry pdf, syllabus.

Example 3: In an Isosceles triangle, the two equal angles are ${40}^{\circ }$ each. What is the measure of the third angle?

Solution:

1. First, let’s understand what we need to find: We know two angles are ${40}^{\circ }$ and ${40}^{\circ }$. The total must be ${180}^{\circ }$.
2. Now, write down the formula: ${40}^{\circ }+{40}^{\circ }+x={180}^{\circ }$
3. Combine the known numbers: ${80}^{\circ }+x={180}^{\circ }$
4. Calculate $x$: Subtract ${80}^{\circ }$ from ${180}^{\circ }$. $$x={180}^{\circ }-{80}^{\circ }$$ $$x={100}^{\circ }$$
5. So, the answer is: The third angle is ${100}^{\circ }$.

Example 4: One exterior angle of a triangle is ${110}^{\circ }$. The interior angle next to it is $x$. Find $x$.

Solution:

1. First, let’s understand the rule: An interior angle and its exterior angle form a straight line. They are supplementary, meaning they add up to ${180}^{\circ }$.
2. Now, write down the formula: Interior Angle + Exterior Angle $={180}^{\circ }$. $$x+{110}^{\circ }={180}^{\circ }$$
3. Put the numbers in: $x={180}^{\circ }-{110}^{\circ }$
4. Calculate it: $$x={70}^{\circ }$$
5. So, the answer is: The interior angle $x$ is ${70}^{\circ }$. This property is crucial for the SSC GD Geometry pdf, exam.

3. Quadrilaterals (Basic Shapes)

A quadrilateral is any closed shape that has four sides and four corners. Think of a window, a book, or a field.

• Examples: Square, Rectangle, Parallelogram, Rhombus.

The most important rule for quadrilaterals:

The sum of all four interior angles in any quadrilateral is always ${360}^{\circ }$.

$$\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C+\mathrm{\angle }D={360}^{\circ }$$

This rule is the foundation for solving quadrilateral problems in your SSC GD Geometry pdf, studies.

4 Solved Examples

Example 1: Three angles of a quadrilateral are ${80}^{\circ }$, ${95}^{\circ }$, and ${110}^{\circ }$. Find the measure of the fourth angle ($x$).

Solution:

1. First, let’s understand the rule: All four angles must add up to ${360}^{\circ }$.
2. Now, write down the formula: ${80}^{\circ }+{95}^{\circ }+{110}^{\circ }+x={360}^{\circ }$
3. Combine the known numbers: $$80+95=175$$ $$175+110={285}^{\circ }$$ $${285}^{\circ }+x={360}^{\circ }$$
4. Calculate $x$: Subtract ${285}^{\circ }$ from ${360}^{\circ }$. $$x={360}^{\circ }-{285}^{\circ }$$ $$x={75}^{\circ }$$
5. So, the answer is: The fourth angle is ${75}^{\circ }$.

Example 2: In a rectangle, what is the sum of all four interior angles?

Solution:

1. First, let’s understand the shape: A rectangle is a type of quadrilateral. All its corners are right angles (${90}^{\circ }$).
2. Now, write down the formula: Sum $={90}^{\circ }+{90}^{\circ }+{90}^{\circ }+{90}^{\circ }$.
3. Calculate it: $$\text{Sum}=4\times {90}^{\circ }$$ $$\text{Sum}={360}^{\circ }$$
4. So, the answer is: The sum of angles in a rectangle is ${360}^{\circ }$. This confirms the general rule for all quadrilaterals.

Example 3: The angles of a quadrilateral are $x$, $2x$, $3x$, and $4x$. Find the value of $x$.

Solution:

1. First, let’s understand what we need to find: The sum of all angles is ${360}^{\circ }$.
2. Now, write down the formula: $x+2x+3x+4x={360}^{\circ }$
3. Combine the terms: $10x={360}^{\circ }$
4. Calculate $x$: Divide ${360}^{\circ }$ by 10. $$x=360/10$$ $$x={36}^{\circ }$$
5. So, the answer is: The value of $x$ is ${36}^{\circ }$. (The angles are ${36}^{\circ },{72}^{\circ },{108}^{\circ },{144}^{\circ }$).

Example 4: A quadrilateral has two right angles (${90}^{\circ }$ each). The third angle is ${100}^{\circ }$. What is the fourth angle?

Solution:

1. First, let’s understand what we need to find: We know three angles: ${90}^{\circ },{90}^{\circ }$, and ${100}^{\circ }$. The total must be ${360}^{\circ }$.
2. Now, write down the formula: ${90}^{\circ }+{90}^{\circ }+{100}^{\circ }+x={360}^{\circ }$
3. Combine the known numbers: $$90+90=180$$ $$180+100={280}^{\circ }$$ $${280}^{\circ }+x={360}^{\circ }$$
4. Calculate $x$: Subtract ${280}^{\circ }$ from ${360}^{\circ }$. $$x={360}^{\circ }-{280}^{\circ }$$ $$x={80}^{\circ }$$
5. So, the answer is: The fourth angle is ${80}^{\circ }$. Keep practicing these types of problems using your SSC GD Geometry pdf, materials!