Q1: Which of the following is the correct notation for "triangle ABC is congruent to triangle DEF"? (a) △ABC ~ △DEF (b) △ABC ≅ △DEF (c) △ABC = △DEF (d) △ABC ≈ △DEF
Solution:
Ans: (b) Explanation: The symbol ≅ denotes congruence, meaning the triangles have the same shape and size. The symbol ~ means similarity, = means equality, and ≈ means approximately equal.
Q2: If △PQR ≅ △STU, which of the following must be true? (a) ∠P ≅ ∠S only (b) PQ ≅ ST and ∠Q ≅ ∠T only (c) All corresponding sides and angles are congruent (d) The triangles have the same perimeter but different angles
Solution:
Ans: (c) Explanation: When two triangles are congruent, all corresponding sides and all corresponding angles are congruent. This is the definition of congruent triangles. Options (a) and (b) are incomplete, and option (d) is incorrect because congruent triangles must have identical angles.
Q3: Which congruence postulate proves that two triangles are congruent if all three corresponding sides are congruent? (a) ASA (Angle-Side-Angle) (b) SAS (Side-Angle-Side) (c) SSS (Side-Side-Side) (d) AAS (Angle-Angle-Side)
Solution:
Ans: (c) Explanation: The SSS (Side-Side-Side) postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. The other postulates require angle information.
Q4: In triangles ABC and XYZ, AB = XY = 5 cm, ∠A ≅ ∠X = 60°, and AC = XZ = 7 cm. Which congruence postulate proves △ABC ≅ △XYZ? (a) SSS (b) SAS (c) ASA (d) AAS
Solution:
Ans: (b) Explanation: We have two sides (AB = XY and AC = XZ) and the included angle between them (∠A ≅ ∠X). This matches the SAS (Side-Angle-Side) postulate. The angle must be between the two given sides for SAS to apply.
Q5: Which set of information is NOT sufficient to prove two triangles congruent? (a) Three pairs of congruent sides (b) Two pairs of congruent angles and one pair of congruent corresponding sides (c) Three pairs of congruent angles (d) Two pairs of congruent sides and the included angle
Solution:
Ans: (c) Explanation: Three pairs of congruent angles (AAA) proves similarity, not congruence. Triangles can have the same angles but different sizes. Options (a), (b), and (d) correspond to SSS, AAS, and SAS, which all prove congruence.
Q6: If △JKL ≅ △MNO with the correspondence J ↔ M, K ↔ N, L ↔ O, and JK = 8 cm, what is the length of MN? (a) Cannot be determined (b) 4 cm (c) 16 cm (d) 8 cm
Solution:
Ans: (d) Explanation: Since △JKL ≅ △MNO with the given correspondence, corresponding sides are congruent. Side JK corresponds to side MN, so MN = JK = 8 cm.
Q7: In △ABC and △DEF, ∠A ≅ ∠D, ∠B ≅ ∠E, and AB ≅ DE. Which congruence postulate can be used? (a) SSS (b) ASA (c) SAS (d) AAS
Solution:
Ans: (b) Explanation: We have two angles (∠A ≅ ∠D and ∠B ≅ ∠E) and the included side between them (AB ≅ DE). This matches the ASA (Angle-Side-Angle) postulate. The side must be between the two given angles for ASA.
Q8: Which statement about congruent figures is true? (a) Congruent figures have the same shape but may have different sizes (b) Congruent figures have the same size but may have different shapes (c) Congruent figures have both the same shape and the same size (d) Congruent figures must be oriented in the same direction
Solution:
Ans: (c) Explanation:Congruent figures have exactly the same shape and size, regardless of position or orientation. Option (a) describes similarity, option (b) is impossible, and option (d) is incorrect because congruent figures can be rotated or reflected.
## Section B: Fill in the Blanks Q9:The symbol __________ is used to indicate that two geometric figures are congruent.
Solution:
Ans: ≅ Explanation: The congruence symbol ≅ is used to show that two figures have the same shape and size with all corresponding parts equal.
Q10:If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent by the __________ postulate.
Solution:
Ans: AAS Explanation: The AAS (Angle-Angle-Side) postulate states that if two angles and a non-included side are congruent, the triangles are congruent.
Q11:In a congruence statement △ABC ≅ △PQR, the angle corresponding to ∠B is __________.
Solution:
Ans: ∠Q Explanation: In congruence statements, corresponding vertices are written in the same order. Since B is the second vertex in △ABC, the corresponding angle is ∠Q, the second vertex in △PQR.
Q12:The __________ postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
Solution:
Ans: SAS Explanation: The SAS (Side-Angle-Side) postulate requires two sides and the angle between them (the included angle) to prove triangle congruence.
Q13:Two triangles with three pairs of congruent angles are __________ but not necessarily congruent.
Solution:
Ans: similar Explanation:AAA (Angle-Angle-Angle) proves that triangles are similar (same shape), but they may have different sizes, so they are not necessarily congruent.
Q14:If △RST ≅ △UVW and the perimeter of △RST is 24 cm, then the perimeter of △UVW is __________ cm.
Solution:
Ans: 24 Explanation:Congruent triangles have all corresponding sides equal, so their perimeters are also equal. If △RST ≅ △UVW, they have the same perimeter.
## Section C: Word Problems Q15:A construction company is building two identical triangular garden plots. The first plot has sides measuring 12 feet, 15 feet, and 18 feet. What are the side lengths of the second plot if it is congruent to the first?
Solution:
Ans: Step 1: Since the two plots are congruent, all corresponding sides are equal. Step 2: The second plot has the same side lengths as the first plot. Final Answer: The second plot has sides measuring 12 feet, 15 feet, and 18 feet.
Q16:Two triangular signs are being made for a school. In △ABC, AB = 8 inches, ∠A = 45°, and AC = 10 inches. In △DEF, DE = 8 inches, ∠D = 45°, and DF = 10 inches. Prove that the two signs are congruent and name the postulate used.
Solution:
Ans: Step 1: Identify the given information: AB = DE = 8 inches, ∠A ≅ ∠D = 45°, AC = DF = 10 inches. Step 2: The angle ∠A (or ∠D) is the included angle between sides AB and AC (or DE and DF). Step 3: Since two sides and the included angle are congruent, we use the SAS postulate. Final Answer: The two signs are congruent by the SAS (Side-Angle-Side) postulate.
Q17:A bridge has two support beams forming triangles. In △PQR, ∠P = 50°, ∠Q = 60°, and PQ = 20 feet. In △XYZ, ∠X = 50°, ∠Y = 60°, and XY = 20 feet. Are the two triangular supports congruent? If so, which postulate proves it?
Solution:
Ans: Step 1: Identify the given: ∠P ≅ ∠X = 50°, ∠Q ≅ ∠Y = 60°, PQ ≅ XY = 20 feet. Step 2: We have two angles and the included side between them (the side connecting the two angles). Step 3: This matches the ASA (Angle-Side-Angle) postulate. Final Answer: Yes, the two triangular supports are congruent by the ASA postulate.
Q18:In △ABC and △DEF, it is known that AB ≅ DE, BC ≅ EF, and AC ≅ DF. If ∠B = 75° in △ABC, what is the measure of the corresponding angle in △DEF?
Solution:
Ans: Step 1: Since AB ≅ DE, BC ≅ EF, and AC ≅ DF, the triangles are congruent by SSS postulate. Step 2: In congruent triangles, all corresponding angles are congruent. Step 3: ∠B corresponds to ∠E based on the order of vertices in the congruence. Final Answer: The measure of ∠E is 75°.
Q19:A designer creates two congruent triangular logos. If one logo has angles measuring 40°, 60°, and 80°, and one side measures 5 cm, can you determine all the side lengths of the second logo? Explain why or why not.
Solution:
Ans: Step 1: Having three angles (AAA) only proves the triangles are similar, not congruent. Step 2: However, we are told the logos are congruent, so all corresponding parts are equal. Step 3: We know one side is 5 cm, but without knowing which side or having additional side information, we cannot determine all three side lengths of the second logo. Final Answer: No, we cannot determine all side lengths. We need to know all three side lengths or which specific side measures 5 cm along with additional congruence information.
Q20:Two triangular sections of a quilt are congruent. If △JKL ≅ △MNO with JK = 6 inches, KL = 8 inches, JL = 10 inches, and ∠K = 90°, find the length of the hypotenuse of △MNO and identify which angle is the right angle.
Solution:
Ans: Step 1: Since △JKL ≅ △MNO, all corresponding parts are congruent. Step 2: In △JKL, the longest side is JL = 10 inches, which is the hypotenuse (opposite the right angle ∠K). Step 3: The corresponding angle to ∠K in △MNO is ∠N (based on vertex order). Step 4: The corresponding side to JL in △MNO is MO = 10 inches (the hypotenuse). Final Answer: The hypotenuse of △MNO is 10 inches, and ∠N is the right angle (90°).
The document Worksheet (with Solutions): Congruence is a part of the Grade 9 Course Integrated Math 1.
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