SUMMARY: This chapter focuses on the properties and criteria for the congruence of triangles. KEY TOPICS: congruence of triangles, criteria for congruence, SSS criterion, SAS criterion, ASA criterion, RHS criterion, inequalities in triangles, properties of triangles, angle sum property, exterior angle theorem
If two triangles have two sides equal and the angle included between those sides is also equal, what criterion can be used to prove their congruence?
ASide-Side-Side (SSS)
BAngle-Side-Angle (ASA)
CSide-Angle-Side (SAS)
DAngle-Angle (AA)
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Correct answer: Option 3 — Side-Angle-Side (SAS)
Q31 Mark
In triangle ABC, if angle A = 50°, angle B = 60°, and angle C = 70°, which property of triangles is illustrated?
AExterior angle theorem
BAngle sum property
CInequalities in triangles
DCongruence criteria
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Correct answer: Option 2 — Angle sum property
Q41 Mark
Which of the following statements is true regarding the RHS criterion for triangle congruence?
AIt requires two sides and the included angle to be equal.
BIt applies only to right-angled triangles.
CIt can be used for any type of triangle.
DIt is equivalent to the SSS criterion.
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Correct answer: Option 2 — It applies only to right-angled triangles.
Q51 Mark
If in triangle ABC, AB = 5 cm, AC = 7 cm, and BC = 10 cm, which inequality is satisfied?
AAB + AC > BC
BAB + BC > AC
CAC + BC > AB
DAll of the above
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Correct answer: Option 4 — All of the above
Short Answer Questions5 questions
Q63 Marks
What is the SSS criterion for the congruence of triangles?
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The SSS (Side-Side-Side) criterion states that if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent.
Q73 Marks
Explain the SAS criterion and provide an example.
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The SAS (Side-Angle-Side) criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. For example, if triangle ABC has AB = 5 cm, AC = 7 cm, and angle A = 60°, and triangle DEF has DE = 5 cm, DF = 7 cm, and angle D = 60°, then triangle ABC is congruent to triangle DEF.
Q83 Marks
What does the ASA criterion for triangle congruence state?
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The ASA (Angle-Side-Angle) criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
Q93 Marks
State the exterior angle theorem and its significance in triangle properties.
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The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. This theorem is significant as it helps in solving for unknown angles in triangles and reinforces the properties of triangle congruence.
Q103 Marks
How do inequalities in triangles relate to the sides and angles of a triangle?
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Inequalities in triangles state that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Additionally, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side, which helps in understanding the relationship between sides and angles in triangles.
Long Answer Questions5 questions
Q116 Marks
Explain the SSS (Side-Side-Side) criterion for the congruence of triangles. Provide an example to illustrate how this criterion can be used to prove that two triangles are congruent.
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The SSS criterion states that if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent. This means that the corresponding angles of the triangles are also equal. For example, consider triangle ABC with sides AB = 5 cm, BC = 7 cm, and AC = 8 cm, and triangle DEF with sides DE = 5 cm, EF = 7 cm, and DF = 8 cm. Since all corresponding sides are equal, by the SSS criterion, triangle ABC is congruent to triangle DEF.
Q126 Marks
Discuss the SAS (Side-Angle-Side) criterion for triangle congruence. How does this criterion differ from the SSS criterion? Provide an example to support your explanation.
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The SAS criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent. This differs from the SSS criterion, which requires all three sides to be equal. For instance, if triangle XYZ has sides XY = 6 cm, YZ = 4 cm, and angle Y = 50 degrees, and triangle PQR has sides PQ = 6 cm, QR = 4 cm, and angle Q = 50 degrees, then by the SAS criterion, triangle XYZ is congruent to triangle PQR as the two sides and the included angle are equal.
Q136 Marks
What is the ASA (Angle-Side-Angle) criterion for triangle congruence? Provide a detailed example to demonstrate how this criterion can be applied in a practical scenario.
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The ASA criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent. For example, consider triangle ABC where angle A = 30 degrees, angle B = 60 degrees, and side AB = 5 cm. Now, consider triangle DEF where angle D = 30 degrees, angle E = 60 degrees, and side DE = 5 cm. Since two angles and the included side are equal in both triangles, by the ASA criterion, triangle ABC is congruent to triangle DEF. This criterion is particularly useful in construction and design, where specific angles and lengths need to be matched.
Q146 Marks
Explain the RHS (Right angle-Hypotenuse-Side) criterion for the congruence of right triangles. How does this criterion simplify the process of proving triangle congruence?
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The RHS criterion states that if the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, then the two triangles are congruent. This criterion simplifies the process of proving triangle congruence because it only requires the measurement of the hypotenuse and one leg, rather than all three sides or angles. For instance, if triangle ABC is a right triangle with hypotenuse AC = 10 cm and one leg AB = 6 cm, and triangle DEF is also a right triangle with hypotenuse DF = 10 cm and one leg DE = 6 cm, then by the RHS criterion, triangle ABC is congruent to triangle DEF.
Q156 Marks
Discuss the angle sum property of triangles and its implications in proving the congruence of triangles. Provide an example to illustrate your explanation.
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The angle sum property of triangles states that the sum of the interior angles of a triangle is always 180 degrees. This property is crucial in proving the congruence of triangles, particularly when using the ASA or AAS (Angle-Angle-Side) criteria. For example, if triangle GHI has angles G = 70 degrees, H = 60 degrees, and I = 50 degrees, and triangle JKL has angles J = 70 degrees, K = 60 degrees, and L = 50 degrees, both triangles have the same angle measures. Therefore, by the AAS criterion, triangle GHI is congruent to triangle JKL. This property also helps in deducing the measures of unknown angles in a triangle, further aiding in the congruence proofs.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): If two triangles are congruent by the SSS criterion, then their corresponding angles are also equal.
Reason (R): The SSS criterion states that if the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The angle sum property of a triangle states that the sum of the angles in a triangle is 180 degrees.
Reason (R): This property is applicable to all types of triangles regardless of their congruence.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): If two angles of one triangle are equal to two angles of another triangle, then the two triangles are congruent.
Reason (R): This statement is true only if the triangles are right-angled, which is a specific case of the ASA criterion.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The RHS criterion can be used to prove the congruence of two right-angled triangles.
Reason (R): The RHS criterion states that if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, then the triangles are congruent.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): In any triangle, the length of one side must be greater than the difference of the lengths of the other two sides.
Reason (R): This is a consequence of the triangle inequality theorem, which is a fundamental property of triangles.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The SSS criterion states that if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent.
Statement 2: The SAS criterion requires that two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle for congruence.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The angle sum property of a triangle states that the sum of the interior angles of a triangle is 180 degrees.
Statement 2: The exterior angle theorem states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The RHS criterion can be used to prove the congruence of two right-angled triangles if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another triangle.
Statement 2: The inequalities in triangles state that the sum of the lengths of any two sides of a triangle must be less than the length of the third side.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q241 Mark
Statement 1: If two angles of a triangle are equal to two angles of another triangle, then the triangles are congruent by the ASA criterion.
Statement 2: The SAS criterion can be used to establish congruence only if the angles are included between the two sides being compared.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: The SSS criterion is applicable only when all three sides of the triangles are known.
Statement 2: The properties of triangles do not include any information about the relationship between the sides and angles.
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Correct answer: Option 2 —
Only Statement 1 is true.
