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1. Median

The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians.

2. Altitude of triangle

3. The perpendicular line segment from a vertex of a triangle to its opposite side iscalled an altitude of the triangle. A triangle has 3 altitudes.

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Difference between median and altitude || triangles || median vs altitude|| class 7

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https://youtu.be/ljwwnk-QeZo?si=DUJwoQS48Cv61cL

Two figures are said to be similar if they have the same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are in the same ratio (proportional).

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10th Class - Similar Figures | Triangles | Tutorials Point

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https://www.youtube.com/live/IFklrv9mLcM?si=5er6QQdry_J4abPX

Theorem 1 (Thales' Theorem) : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other sides are divided in the same ratio.

Theorem 2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

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Similarity of triangles | Triangles | Chapter 7 | Class 10 | Boards 2024 | SHIKHAR | BYJU’S

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Hello Students!!!Get ready to ace every subject with BYJU’S Classes 9&10, a comprehensive education platform exclusively for Classes 9 and 10 Students. The c…

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https://youtu.be/SkmrpY0bXPI?si=MvF9u6_geNyVW04t

Theorem 3 : If in two triangles, corresponding angles are equal, the their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

Theorem 4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

Theorem 5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

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Triangles 02 | All Similarity Criteria in 1 Shot | Theorems A.A.A • S.S.S • S.A.S  Proofs | Class 10

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https://youtu.be/bz-WHDvZCcc?si=tqylDtlLkxTFN4OZ

Theorem 6 : The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

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Areas of Similar Triangles (Theorem 6.6) - Triangles | Class 10 Maths 2023-24

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https://youtu.be/enMIKg5Okrs?si=ImZNDsKiN6jdfWYD

Theorem 7 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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Class 10 Maths Chapter 6 | Pythagoras Theorem Proof - Triangles 2022-23

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Theorem 8 : In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

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Class 10 Maths Chapter 6 | Converse of Pythagoras Theorem Proof - Triangles 2022-23

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A polygon with three sides, three angles, and three vertices is called a triangle. The emphasis is on triangle similarity, where two triangles are said to be similar if their respective sides are proportionate and their corresponding angles are equal. AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) are the three main similarity criterion. Important theorems that aid in the solution of geometric issues include Pythagoras' Theorem and its converse, as well as Thales' Theorem (Basic Proportionality Theorem). The square of the ratio of matching sides determines the area of similar triangles. These ideas are useful for measuring heights and distances in physics, surveying, building, and navigation

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Triangles quiz for 10th grade students. Find other quizzes for Mathematics and more on Quizizz for free!

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Fixed Shape – Cannot change form without altering its properties.

Limited Area – Has a smaller enclosed area compared to other polygons.

Dependence on Measurements – Small errors in sides or angles affect accuracy.

Difficult Constructions – Some triangle properties require complex calculations.

Rigid Structure – Unlike quadrilaterals, they cannot adapt flexibly in designs.

Limited Practical Applications – Not as versatile as some other geometric figures.

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Class 5th-10th Triangles

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https://www.cuemath.com/geometry/exterior-angle-of-triangle/

An exterior angle of a triangle is equal to the sum of its interior oppositeangles

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Congruence of Triangle

Two triangles are congruent if they have the same shape and size. This means that all three sides and all three angles of one triangle are equal to the other triangle.

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Congruence of Triangles | Part 1/3 | English | Class 9

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Two triangles are considered congruent if

all three angles have the same value and

all three sides have the same value.

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Congruence of Triangles | Part 2/3 | English | Class 9

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The total measure of

the three angles of a

triangle is 180°.

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sum of the three angles  of a triangle - for sales contact 9677551115 - (Ramnad  Saravanan)

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Congruent triangles with 3 pieces of information (not the whole 6).  This is because once we know 3 pieces we can use other geometrical angle rules or trigonometry side length rules to calculate the others.

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Congruence of Triangles | Part 3/3 | English | Class 9

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If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the two triangles are congruent, a concept known as the Side-Angle-Side (SAS) congruence criterion.

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SUM OF THE LENGTHS OF TWO SIDES OF A TRIANGLE

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SUM OF THE LENGTHS OF TWO SIDES OF A TRIANGLE

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The sum of the lengths of any two sides of a triangle is greater than the third side.

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Sum of the lengths of any 2 sides of triangle is greater than the third side|Part1/3|English|Class 7

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SAS, ASA, and AAS are congruence postulates used to prove that two triangles are congruent. SAS (Side-Angle-Side) requires two sides and the included angle to be congruent, ASA (Angle-Side-Angle) requires two angles and the included side, and AAS (Angle-Angle-Side) requires two angles and a non-included side to be congruent.

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9.7.3 Congruent Triangles SAS ASA AAS - Triangles - Class 9 CBSE Math3 SAS ASA AAS

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Some more criteria for congruence of triangles

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Some more criteria for congruence of triangles

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Some More Criteria for Congruence of Triangles

You have seen earlier in this chapter that equality of three angles of one triangle to three angles of the other is not sufficient for the congruence of the two triangles. You may wonder whether equality of three sides of one triangle to three sides of another triangle is enough for congruence of the two triangles. You have already verified in earlier classes that this is indeed true.

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RIGHT-ANGLED TRIANGLES AND PYTHAGORAS PROPERTY

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RIGHT-ANGLED TRIANGLES AND PYTHAGORAS PROPERTY

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1.In a right-angled triangle,

the square on the hypotenuse = sum of the squares on the legs

2.If the Pythagoras property holds, the triangle must be right-angled.

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Right-angled Triangles and Pythagoras Property | Shaalaa.com

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Shaalaa.com provides free Question papers and solutions for ICSE, CBSE, Maharashtra Board and university exams with Free PDF downloads. It is a one stop solution to score more in exams.

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Theorem: Angles opposite to the equal sides of an isosceles triangle are also equal.

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Some Properties of a Triangle | Part 1/3 | English | Class 9

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the Side-Side-Side (SSS) congruence rule states that two triangles are congruent if all three sides of one triangle are equal to the corresponding sides of the other triangle.

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Theorem:7.4 (SSS Congruence Rule) Chapter:7 Triangles | Ncert Maths Class 9  MOST IMPORTANT THEOREM

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The RHS (Right Angle-Hypotenuse-Side) congruence rule states that two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the corresponding hypotenuse and side of the other triangle.

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RHS Congruence Rule - Theorem 7.5 Class 9 [with Proof and Example]  - Triangles

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Two triangles are congruent if they have the same size and shape, meaning all three corresponding sides and all three corresponding angles are equal.

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SAS congruence criterion:If under a correspondence, two sides and the angle included between them of a triangle

are equal to two corresponding sides and the angle included between them of another

triangle, then the triangles are congruent.

2.ASA Congruence criterion:

If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.

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Congruency Rules - SAS And ASA | Class 7 | Learn With BYJU'S

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RHS Congruence criterion:

If under a correspondence, the hypotenuse and one side of a right-angled triangle are

respectively equal to the hypotenuse and one side of another right-angled triangle, then

the triangles are congruent.

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Right angle-Hypotenuse- Side (RHS) Congruence Criterion of triangles | Part1/3 | English | Class 7

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NCERT Book Class 10 Maths Chapter 6 Triangles

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a) Types of Triangles (Based on Sides)

1. Equilateral triangle (all sides equal)

2. Isosceles triangle (two sides equal)

3. Scalene triangle (all sides different)

b) Types of Triangles (Based on Angles)

1. Acute-angled triangle (all angles < 90

2. Right-angled triangle (one angle = 90°)

3. Obtuse-angled triangle (one angle > 90°)

Triangle Properties

The sum of the three angles of a triangle is always 180°

Triangle inequality theorem (sum of any two sides is greater than the third side)

Perimeter and Area of a Triangle

Formula for the perimeter (sum of all three sides)

Area of a triangle using 1/2 × base × height

Constructing Triangles

Constructing triangles using a compass and ruler

Drawing triangles with given sides and angles

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Types of Triangles - Definitions, Properties, Examples

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1.Constructing a triangle when the lengths of two sides and the measure of the angle between them are known. (SAS criterion)

2.Constructing a triangle when the measures of two of its angles and the length of the side included between them is given. (ASA criterion)

3.Constructing a right-angled triangle when the length of one leg and its hypotenuse are given (RHS criterion)

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Triangle Constructions: SSS, SAS, ASA | Addvance Maths 😊➕

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1. A figure has three sides and three angles. Is it always a triangle? Why?

Yes, because a triangle is defined as a closed shape with three sides and three angles.

2. Identify the type of triangle if:

a) All three sides are equal to 6 cm.

b) Two sides are 5 cm, and the third side is 8 cm.

c) The sides measure 3 cm, 4 cm, and 5 cm.

a) Equilateral triangle

b) Isosceles triangle

c) Scalene triangle

3. Classify the triangles based on angles:

a) A triangle with angles 50°, 60°, and 70°.

b) A triangle with angles 30°, 60°, and 90°.

c) A triangle with angles 30°, 120°, and 30°.

a) Acute-angled triangle

b) Right-angled triangle

c) Obtuse-angled triangle

4. Find the missing angle in a triangle if the other two angles are 65° and 75°.

Missing angle = 180° - (65° + 75°) = 40°

5. Find the perimeter of a triangle with sides 7 cm, 10 cm, and 5 cm.

Perimeter = 7 + 10 + 5 = 22 cm

6. Find the area of a triangle with base = 8 cm and height = 5 cm.

Area = (1/2) × 8 × 5 = 20 cm²

7. Can a triangle have sides 3 cm, 4 cm, and 8 cm?

No, because 3 + 4 = 7, which is less than 8. This violates the triangle inequality theorem.

8. Construct a triangle with sides 5 cm, 6 cm, and 7 cm. What steps would you follow?

Draw a base of 7 cm, use a compass to draw arcs of 5 cm and 6 cm, and join the intersection point to complete the triangle.

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A triangle is a three-sided polygon with three angles and three vertices. It is the simplest closed figure in geometry and plays a fundamental role in mathematics. Triangles are classified based on sides (scalene, isosceles, equilateral) and angles (acute, right, obtuse). The angle sum property states that the sum of all interior angles of a triangle is always 180°. Pythagoras' theorem applies to right-angled triangles, helping in distance calculations. Congruence and similarity of triangles form the basis of many geometric proofs. Triangles are used in architecture, construction, and trigonometry. They also appear in navigation, engineering, and various scientific applications.

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Fundamental Geometric Shape – Forms the base of geometry.

Structural Strength – Used in bridges and buildings for stability.

Mathematical Applications – Key in trigonometry, geometry, and calculus.

Real-World Usage – Found in art, design, and mechanical structures.

Navigation & Physics – Used in GPS, astronomy, and force analysis.

Foundation for Higher Studies – Essential for advanced mathematics and engineering.

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Understand the classification of triangles based on sides and angles.

Learn and apply the angle sum property and exterior angle property.

Understand and use the triangle inequality theorem.

Apply congruence criteria (SSS, SAS, ASA, RHS) to solve problems.

Understand similarity criteria (AA, SSS, SAS) and their applications.

Prove and apply Thales’ theorem and Pythagoras theorem.

Solve mathematical problems using triangle properties and theorems.

Apply triangle concepts in real-life situations like height and distance calculations.