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Questions and Answers
What is the definition of similar triangles?
Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
What is the AAA similarity criterion?
If two triangles have three equal angles, they are similar.
What is the SSS congruence criterion?
If two triangles have three equal sides, they are congruent.
What is the sum of angles in a triangle?
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What is the exterior angle of a triangle equal to?
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What is the Pythagoras' theorem?
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What is the relationship between the longest side and the largest angle in a triangle?
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What is the RHS congruence criterion?
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What is the inequality relation between the sides of a triangle?
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What is the AA similarity criterion?
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Study Notes
Triangles
Similarity of Triangles
- Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
- If two triangles are similar, their corresponding sides are in proportion.
- AAA (Angle-Angle-Angle) similarity criterion: If two triangles have three equal angles, they are similar.
- AA (Angle-Angle) similarity criterion: If two triangles have two equal angles, they are similar.
- SSS (Side-Side-Side) similarity criterion: If two triangles have three proportional sides, they are similar.
- SAS (Side-Angle-Side) similarity criterion: If two triangles have two proportional sides and the included angle is equal, they are similar.
Congruence of Triangles
- Two triangles are congruent if they have the same shape and size.
- SSS (Side-Side-Side) congruence criterion: If two triangles have three equal sides, they are congruent.
- SAS (Side-Angle-Side) congruence criterion: If two triangles have two equal sides and the included angle is equal, they are congruent.
- ASA (Angle-Side-Angle) congruence criterion: If two triangles have two equal angles and the included side is equal, they are congruent.
- AAS (Angle-Angle-Side) congruence criterion: If two triangles have two equal angles and one equal side, they are congruent.
- RHS (Right Angle-Hypotenuse-Side) congruence criterion: If two right-angled triangles have equal hypotenuse and one equal side, they are congruent.
Properties of Triangles
- Sum of angles in a triangle is 180°.
- Exterior angle of a triangle is equal to the sum of the two interior opposite angles.
- Angle opposite to the longest side is the largest angle.
- Side opposite to the largest angle is the longest side.
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras' theorem).
Inequalities in Triangles
- Sum of any two sides of a triangle is greater than the third side.
- Difference of any two sides of a triangle is less than the third side.
- Triangle inequality theorem: If a, b, and c are the sides of a triangle, then |a-b| < c < a+b.
Triangles
Similarity of Triangles
- Corresponding angles of similar triangles are equal.
- Corresponding sides of similar triangles are proportional.
- AAA criterion: Three equal angles imply similar triangles.
- AA criterion: Two equal angles imply similar triangles.
- SSS criterion: Three proportional sides imply similar triangles.
- SAS criterion: Two proportional sides and one equal included angle imply similar triangles.
Congruence of Triangles
- Congruent triangles have the same shape and size.
- SSS criterion: Three equal sides imply congruent triangles.
- SAS criterion: Two equal sides and one equal included angle imply congruent triangles.
- ASA criterion: Two equal angles and one equal included side imply congruent triangles.
- AAS criterion: Two equal angles and one equal side imply congruent triangles.
- RHS criterion: Equal hypotenuse and one equal side in right-angled triangles imply congruent triangles.
Properties of Triangles
- The sum of angles in a triangle is always 180°.
- An exterior angle is equal to the sum of its two interior opposite angles.
- The largest angle is opposite to the longest side.
- The longest side is opposite to the largest angle.
- Pythagoras' theorem: a² + b² = c² in right-angled triangles.
Inequalities in Triangles
- The sum of any two sides is greater than the third side.
- The difference of any two sides is less than the third side.
- Triangle inequality theorem: |a-b| < c < a+b for sides a, b, and c.
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Description
Learn about the rules and criteria for determining similar triangles, including the AAA, AA, and SSS similarity criteria.
