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Isosceles and Equilateral Triangle Fun: Easy Theorems and Worksheets for Class 9

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Isosceles and Equilateral Triangle Fun: Easy Theorems and Worksheets for Class 9
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Katie Whitson

@katie_.w0843

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458 Followers

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The isosceles triangle theorem and its converse are fundamental concepts in geometry, explaining the relationship between sides and angles in isosceles triangles. This guide covers these theorems, their applications, and related concepts for equilateral triangles.

Key points:

  • An isosceles triangle has at least two congruent sides
  • The isosceles triangle theorem states that angles opposite congruent sides are congruent
  • The converse theorem states that if two angles are congruent, the opposite sides are congruent
  • Equilateral triangles are both equilateral and equiangular

Highlight: Understanding these theorems is crucial for solving geometry problems involving isosceles and equilateral triangles.

9/21/2023

59

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

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Applying Isosceles Triangle Theorems

This page demonstrates how to apply the isosceles triangle theorems in various geometric problems.

Example A: Determining if sides are congruent based on angle measurements Given: ∠ABC ≅ ∠ACB Question: Is AB congruent to CB? Answer: Yes, according to the Converse of the Isosceles Triangle Theorem, if two angles are congruent, the opposite sides are also congruent.

Example B: Determining if angles are congruent based on side measurements Given: AD ≅ DE Question: Is ∠A congruent to ∠DEA? Answer: Yes, according to the Isosceles Triangle Theorem, if two sides are congruent, the angles opposite those sides are also congruent.

Example: In a triangle WVS, if ∠WVS ≅ ∠S and TR ≅ TS, we can conclude that ∠W ≅ ∠S (by Isosceles Triangle Theorem) and TR ≅ TS (by Converse of Isosceles Triangle Theorem).

Theorem 22: Angle Bisector Theorem for Isosceles Triangles This theorem states that if a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.

In mathematical notation: If AC ≅ BC and ∠ACD ≅ ∠BCD, then CD ⊥ AB and AD ≅ BD

Highlight: The angle bisector theorem for isosceles triangles combines the concepts of angle bisectors and perpendicular bisectors, making it a powerful tool for solving complex geometry problems.

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

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Algebraic Applications and Equilateral Triangles

This page covers algebraic applications in isosceles triangles and introduces properties of equilateral triangles.

Algebraic Applications in Isosceles Triangles

Example 1: Finding the value of x in an isosceles triangle Given: An isosceles triangle with base angles measuring x+5° each and a vertex angle of 90° Solution: Using the angle sum theorem for triangles (180°) and the isosceles triangle theorem: (x+5°) + (x+5°) + 90° = 180° 2x + 100° = 180° 2x = 80° x = 40°

Example: In an isosceles triangle, if one base angle is represented as 4x° and the other as 42°, we can set up the equation 4x° = 42° to solve for x.

Corollary to Theorem 20: Equilateral Triangle Property If a triangle is equilateral (all sides congruent), then it is also equiangular (all angles congruent and measure 60°).

Corollary to Theorem 21: Equiangular Triangle Property If a triangle is equiangular (all angles congruent and measure 60°), then it is also equilateral (all sides congruent).

Highlight: The properties of equilateral triangle include both equal sides and equal angles, making them a special case of isosceles triangles with additional symmetry.

Vocabulary: A corollary is a statement that follows directly from a theorem or definition.

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

View

Isosceles and Equilateral Triangles

This page introduces the concept of isosceles triangles and presents two important theorems related to them.

An isosceles triangle is defined as a triangle with at least two congruent sides. The key components of an isosceles triangle are:

  • Vertex angle: The angle formed by the two congruent sides
  • Legs: The two congruent sides
  • Base: The side opposite the vertex angle
  • Base angles: The angles adjacent to the base

Vocabulary: Congruent means equal in measure or size.

Theorem 20: Isosceles Triangle Theorem This theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. In mathematical notation: If AC ≅ BC, then ∠A ≅ ∠B

Example: In an isosceles triangle ABC with AC = BC, the angles opposite these sides (∠A and ∠B) will be equal.

Theorem 21: Converse of the Isosceles Triangle Theorem This theorem is the reverse of Theorem 20. It states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. In mathematical notation: If ∠A ≅ ∠B, then AC ≅ BC

Highlight: The converse of isosceles triangle theorem is crucial for proving that a triangle is isosceles based on angle measurements.

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Classifying Triangles

How do we classify triangle by examine their angles? How about by their known sides? How do we solve triangle side problems involving isosceles and equilateral triangles?

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Subjects

/

Geometry

/

Congruent Triangles

/

Working with Triangles

Isosceles and Equilateral Triangle Fun: Easy Theorems and Worksheets for Class 9

user profile picture

Katie Whitson

@katie_.w0843

·

458 Followers

Follow

The isosceles triangle theorem and its converse are fundamental concepts in geometry, explaining the relationship between sides and angles in isosceles triangles. This guide covers these theorems, their applications, and related concepts for equilateral triangles.

Key points:

  • An isosceles triangle has at least two congruent sides
  • The isosceles triangle theorem states that angles opposite congruent sides are congruent
  • The converse theorem states that if two angles are congruent, the opposite sides are congruent
  • Equilateral triangles are both equilateral and equiangular

Highlight: Understanding these theorems is crucial for solving geometry problems involving isosceles and equilateral triangles.

9/21/2023

59

 

10th

 

Geometry

3

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

Applying Isosceles Triangle Theorems

This page demonstrates how to apply the isosceles triangle theorems in various geometric problems.

Example A: Determining if sides are congruent based on angle measurements Given: ∠ABC ≅ ∠ACB Question: Is AB congruent to CB? Answer: Yes, according to the Converse of the Isosceles Triangle Theorem, if two angles are congruent, the opposite sides are also congruent.

Example B: Determining if angles are congruent based on side measurements Given: AD ≅ DE Question: Is ∠A congruent to ∠DEA? Answer: Yes, according to the Isosceles Triangle Theorem, if two sides are congruent, the angles opposite those sides are also congruent.

Example: In a triangle WVS, if ∠WVS ≅ ∠S and TR ≅ TS, we can conclude that ∠W ≅ ∠S (by Isosceles Triangle Theorem) and TR ≅ TS (by Converse of Isosceles Triangle Theorem).

Theorem 22: Angle Bisector Theorem for Isosceles Triangles This theorem states that if a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.

In mathematical notation: If AC ≅ BC and ∠ACD ≅ ∠BCD, then CD ⊥ AB and AD ≅ BD

Highlight: The angle bisector theorem for isosceles triangles combines the concepts of angle bisectors and perpendicular bisectors, making it a powerful tool for solving complex geometry problems.

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

Algebraic Applications and Equilateral Triangles

This page covers algebraic applications in isosceles triangles and introduces properties of equilateral triangles.

Algebraic Applications in Isosceles Triangles

Example 1: Finding the value of x in an isosceles triangle Given: An isosceles triangle with base angles measuring x+5° each and a vertex angle of 90° Solution: Using the angle sum theorem for triangles (180°) and the isosceles triangle theorem: (x+5°) + (x+5°) + 90° = 180° 2x + 100° = 180° 2x = 80° x = 40°

Example: In an isosceles triangle, if one base angle is represented as 4x° and the other as 42°, we can set up the equation 4x° = 42° to solve for x.

Corollary to Theorem 20: Equilateral Triangle Property If a triangle is equilateral (all sides congruent), then it is also equiangular (all angles congruent and measure 60°).

Corollary to Theorem 21: Equiangular Triangle Property If a triangle is equiangular (all angles congruent and measure 60°), then it is also equilateral (all sides congruent).

Highlight: The properties of equilateral triangle include both equal sides and equal angles, making them a special case of isosceles triangles with additional symmetry.

Vocabulary: A corollary is a statement that follows directly from a theorem or definition.

3.5 Isoceles & Equilateral Triangles Isoceles Triangles- with at least 2 cong. sides. vertex angle Legs Base Base Angles Theorem 20-Isoceles

Isosceles and Equilateral Triangles

This page introduces the concept of isosceles triangles and presents two important theorems related to them.

An isosceles triangle is defined as a triangle with at least two congruent sides. The key components of an isosceles triangle are:

  • Vertex angle: The angle formed by the two congruent sides
  • Legs: The two congruent sides
  • Base: The side opposite the vertex angle
  • Base angles: The angles adjacent to the base

Vocabulary: Congruent means equal in measure or size.

Theorem 20: Isosceles Triangle Theorem This theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. In mathematical notation: If AC ≅ BC, then ∠A ≅ ∠B

Example: In an isosceles triangle ABC with AC = BC, the angles opposite these sides (∠A and ∠B) will be equal.

Theorem 21: Converse of the Isosceles Triangle Theorem This theorem is the reverse of Theorem 20. It states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. In mathematical notation: If ∠A ≅ ∠B, then AC ≅ BC

Highlight: The converse of isosceles triangle theorem is crucial for proving that a triangle is isosceles based on angle measurements.

Similar Content

Geometry - Classifying Triangles

How do we classify triangle by examine their angles? How about by their known sides? How do we solve triangle side problems involving isosceles and equilateral triangles?

101

423

0

Know Classifying Triangles thumbnail

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the # 1 ranked education app in five European countries

4.9+

Average App Rating

15 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying