SUMMARY: The chapter "Congruence of Triangles" focuses on understanding the concept of congruence in triangles and the criteria for triangle congruence. KEY TOPICS: congruence, congruent figures, criteria for congruence, SSS criterion, SAS criterion, ASA criterion, RHS criterion, properties of congruent triangles, applications of congruence, examples and exercises
Which of the following is a criterion for proving that two triangles are congruent?
AAngle-Side-Angle (ASA)
BSide-Angle-Side (SAS)
CSide-Side-Side (SSS)
DAll of the above
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Correct answer: Option 4 — All of the above
Q21 Mark
If two triangles have two sides equal and the included angle equal, what criterion can be used to prove the triangles are congruent?
AAngle-Angle (AA)
BSide-Side-Angle (SSA)
CSide-Angle-Side (SAS)
DAngle-Side-Angle (ASA)
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Correct answer: Option 3 — Side-Angle-Side (SAS)
Q31 Mark
Which property is true for congruent triangles?
ATheir corresponding angles are equal.
BTheir corresponding sides are unequal.
CThey have different areas.
DTheir corresponding sides are proportional.
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Correct answer: Option 1 — Their corresponding angles are equal.
Q41 Mark
In triangle ABC and triangle DEF, if AB = DE, BC = EF, and AC = DF, which criterion can be applied?
AAngle-Side-Angle (ASA)
BSide-Side-Side (SSS)
CAngle-Angle-Side (AAS)
DSide-Angle-Side (SAS)
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Correct answer: Option 2 — Side-Side-Side (SSS)
Q51 Mark
The Right Angle-Hypotenuse-Side (RHS) criterion is applicable to which type of triangles?
AIsosceles triangles
BEquilateral triangles
CRight-angled triangles
DScalene triangles
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Correct answer: Option 3 — Right-angled triangles
Short Answer Questions5 questions
Q63 Marks
What are congruent figures? Provide an example of congruent triangles.
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Congruent figures are figures that have the same shape and size. For example, if triangle ABC is congruent to triangle DEF, then all corresponding sides and angles of both triangles are equal.
Q73 Marks
Explain the SSS criterion for triangle congruence.
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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 will also be equal.
Q83 Marks
What is the SAS criterion? Provide a brief example to illustrate it.
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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. For example, if triangle ABC has sides AB = 5 cm, AC = 7 cm and angle A = 60°, and triangle DEF has sides DE = 5 cm, DF = 7 cm and angle D = 60°, then triangle ABC is congruent to triangle DEF.
Q93 Marks
Describe the ASA criterion for triangle congruence and its significance.
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The ASA criterion states that if two angles and the side between them in one triangle are equal to two angles and the side between them in another triangle, then the triangles are congruent. This is significant because it allows us to establish congruence without needing to know the lengths of all sides.
Q103 Marks
What is the RHS criterion for triangle congruence? Give an example of its application.
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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. For example, if triangle ABC and triangle DEF are both right triangles with hypotenuse AB = 10 cm and one side AC = 6 cm equal to DE = 10 cm and DF = 6 cm, then triangle ABC is congruent to triangle DEF.
Long Answer Questions5 questions
Q116 Marks
Explain the SSS (Side-Side-Side) criterion for the congruence of triangles. Provide a detailed 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 all corresponding angles are also equal. For example, consider triangle ABC with sides AB = 5 cm, BC = 7 cm, and AC = 8 cm. If we have another triangle DEF with sides DE = 5 cm, EF = 7 cm, and DF = 8 cm, we can conclude that triangle ABC is congruent to triangle DEF by the SSS criterion. This congruence implies that the angles A, B, and C are equal to angles D, E, and F respectively.
Q126 Marks
Describe the ASA (Angle-Side-Angle) criterion for triangle congruence. Illustrate your explanation with a suitable diagram and an example.
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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 instance, if we have triangle PQR where angle P = 50 degrees, angle Q = 60 degrees, and side PQ = 10 cm, and triangle XYZ where angle X = 50 degrees, angle Y = 60 degrees, and side XY = 10 cm, we can conclude that triangle PQR is congruent to triangle XYZ by the ASA criterion. A diagram would show both triangles with the corresponding angles and sides marked to visualize the congruence.
Q136 Marks
What is the RHS (Right angle-Hypotenuse-Side) criterion for the congruence of right-angled triangles? Provide a detailed example to demonstrate this criterion.
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The RHS criterion states that if in two right-angled triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. For example, consider right triangle ABC where angle C is 90 degrees, AC (hypotenuse) = 13 cm, and BC = 5 cm. Now, if we have another right triangle DEF where angle F is 90 degrees, DE (hypotenuse) = 13 cm, and EF = 5 cm, we can conclude that triangle ABC is congruent to triangle DEF by the RHS criterion. This means all corresponding angles and sides are equal.
Q146 Marks
Discuss the properties of congruent triangles and how they can be applied in real-life situations. Provide at least two examples.
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Congruent triangles have several properties, including that their corresponding sides are equal in length and their corresponding angles are equal in measure. These properties can be applied in various real-life situations. For instance, in construction, ensuring that two triangular supports are congruent can guarantee that they will bear the same load. Another example is in art, where congruent triangles can be used to create symmetrical designs. By applying the properties of congruence, artists can ensure that their work is balanced and aesthetically pleasing.
Q156 Marks
Explain the SAS (Side-Angle-Side) criterion for triangle congruence. Provide an example with a diagram 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. For example, consider triangle JKL where JK = 6 cm, KL = 8 cm, and angle K = 45 degrees. If we have another triangle MNO where MN = 6 cm, NO = 8 cm, and angle N = 45 degrees, we can conclude that triangle JKL is congruent to triangle MNO by the SAS criterion. A diagram would show both triangles with the corresponding sides and angles marked, demonstrating their congruence.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): If two triangles are congruent, then their corresponding sides are equal.
Reason (R): The SSS criterion states that if three sides of one triangle are equal to 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 ASA criterion can be used to prove that two triangles are congruent.
Reason (R): 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 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.
Q181 Mark
Assertion (A): If two triangles have two sides equal and the included angle of one triangle is greater than the included angle of the other, then the triangles are congruent.
Reason (R): 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 triangles are congruent.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The RHS criterion is applicable only for 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): Congruent triangles can have different shapes but must have the same size.
Reason (R): Congruent figures are identical in shape and size.
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Correct answer: Option 4 —
A is false, but R is true.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: Two triangles are congruent if all their corresponding sides are equal.
Statement 2: The SSS 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.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: If two angles of one triangle are equal to two angles of another triangle, then the triangles are congruent.
Statement 2: The ASA criterion can be used to prove the congruence of triangles.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The RHS criterion is applicable only for right-angled triangles.
Statement 2: Congruent triangles can have different shapes but the same area.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q241 Mark
Statement 1: If two triangles are congruent, then their corresponding altitudes are also equal.
Statement 2: The SAS criterion requires that two sides and the angle opposite one of them are equal.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: Congruent figures have the same shape and size.
Statement 2: The properties of congruent triangles include that corresponding angles are equal and corresponding sides are proportional.
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Correct answer: Option 4 —
Both statements are false.
