What Are Congruent Figures?

Congruent figures are shapes that are identical in size and shape. They can be flipped, rotated, or moved, but they still remain congruent if they have:

• The same size.
• The same shape.
• Corresponding sides that are equal.
• Corresponding angles that are equal.

Congruence is an important concept in geometry and is often used to explore relationships between shapes, especially triangles and other polygons.

Key Properties of Congruent Figures:

1. Side Lengths of Congruent Figures:
   The side lengths of congruent figures are always equal. For example, if two triangles are congruent, the corresponding sides in both triangles will have the same measurement.
2. Angle Measures of Congruent Figures:
   The corresponding angles of congruent figures are also equal. If two polygons are congruent, their corresponding angles will measure the same.
3. Notation for Congruence:
   Congruence is represented using the symbol ≅. For example, if two triangles ABC and DEF are congruent, we write:
   $$\triangle ABC \cong \triangle DEF$$
   This notation tells us that all corresponding sides and angles of the two triangles are equal.

Practice Questions: Side Lengths and Angle Measures of Congruent Figures

Easy Level Questions

1. If triangle ABC is congruent to triangle DEF, and the side AB = 6 cm, what is the length of side DE?
2. In two congruent squares, one side of the first square is 8 cm. What is the length of a side in the second square?
3. Triangle XYZ is congruent to triangle PQR. If angle ∠X = 40°, what is the measure of angle ∠P?
4. If two congruent rectangles have a side length of 12 cm, what is the length of the corresponding side in the other rectangle?
5. Two congruent triangles have angles of 50°, 60°, and 70°. What are the angles in the other triangle?
6. If triangle ABC is congruent to triangle XYZ and side AC = 7 cm, what is the length of side XZ?
7. If quadrilateral PQRS is congruent to quadrilateral ABCD, and angle ∠P = 90°, what is the measure of angle ∠A?
8. Two congruent pentagons have a side length of 5 cm. What is the side length of the other pentagon?
9. If triangle DEF is congruent to triangle GHI, and angle ∠D = 80°, what is the measure of angle ∠G?
10. If two congruent parallelograms have side lengths of 10 cm and 14 cm, what are the side lengths of the other parallelogram?
11. Triangle ABC is congruent to triangle XYZ, and the angle ∠B = 45°. What is the measure of angle ∠Y?
12. If two congruent triangles have a side length of 9 cm, what is the corresponding side length in the other triangle?
13. If two congruent pentagons have an angle of 108°, what is the corresponding angle in the second pentagon?
14. In congruent rhombuses, if one side is 7 cm, what is the length of the corresponding side in the other rhombus?
15. Two congruent hexagons have a side length of 4 cm. What is the side length of the second hexagon?
16. Triangle ABC is congruent to triangle DEF, and side BC = 8 cm. What is the length of side EF?
17. Two congruent trapeziums have a base of 15 cm. What is the corresponding base length in the other trapezium?
18. Triangle PQR is congruent to triangle STU. If angle ∠R = 30°, what is the measure of angle ∠U?
19. Two congruent octagons have an interior angle of 135°. What is the corresponding angle in the other octagon?
20. If triangle ABC is congruent to triangle DEF, and side AB = 10 cm, what is the length of side DE?

Medium Level Questions

1. If triangle ABC is congruent to triangle XYZ and angle ∠A = 50°, angle ∠B = 70°, what is the measure of angle ∠C?
2. Two congruent parallelograms have angles of 60° and 120°. What are the angles in the other parallelogram?
3. If triangle PQR is congruent to triangle STU, and side PQ = 13 cm, PR = 15 cm, what is the length of side SU?
4. In two congruent trapeziums, one base measures 18 cm and the other 12 cm. What are the corresponding base lengths in the other trapezium?
5. If triangle ABC is congruent to triangle DEF and side AC = 11 cm, AB = 9 cm, what is the length of sides DF and DE?
6. Two congruent quadrilaterals have angles of 90°, 45°, 135°, and 90°. What are the angles in the second quadrilateral?
7. If triangle XYZ is congruent to triangle PQR and side XY = 14 cm, what is the length of side PQ?
8. In two congruent rhombuses, one diagonal measures 10 cm. What is the length of the corresponding diagonal in the second rhombus?
9. Two congruent pentagons have side lengths of 7 cm. What are the side lengths in the second pentagon?
10. In two congruent triangles, angle ∠A = 45°, angle ∠B = 55°. What is the measure of angle ∠C in both triangles?
11. If two congruent trapeziums have one base measuring 20 cm, what is the corresponding base length in the other trapezium?
12. Triangle DEF is congruent to triangle XYZ, and angle ∠D = 35°. What is the measure of angle ∠X?
13. Two congruent pentagons have side lengths of 8 cm. What is the side length of the second pentagon?
14. In congruent rhombuses, if the diagonals measure 12 cm and 16 cm, what are the diagonal lengths in the second rhombus?
15. Two congruent triangles have sides measuring 5 cm, 6 cm, and 7 cm. What are the side lengths in the second triangle?
16. If two congruent trapeziums have angles of 60° and 120°, what are the corresponding angles in the other trapezium?
17. Triangle ABC is congruent to triangle XYZ, and angle ∠A = 50°, angle ∠B = 40°. What is the measure of angle ∠C?
18. Two congruent hexagons have side lengths of 10 cm. What is the side length of the second hexagon?
19. If triangle DEF is congruent to triangle GHI and side DE = 9 cm, what is the length of side GH?
20. Two congruent squares have a side length of 15 cm. What is the side length of the second square?

Hard Level Questions

1. If two congruent trapeziums have angles of 70°, 110°, 70°, and 110°, what are the angles in the other trapezium?
2. In congruent triangles, the angles measure 40°, 70°, and 70°. What are the angles in the second triangle?
3. If triangle ABC is congruent to triangle DEF and side AB = 12 cm, AC = 15 cm, what is the length of sides DE and DF?
4. Two congruent pentagons have interior angles of 108°. What are the corresponding angles in the second pentagon?
5. Triangle XYZ is congruent to triangle PQR. If side XY = 14 cm and side YZ = 9 cm, what are the corresponding side lengths in triangle PQR?
6. If triangle ABC is congruent to triangle DEF, and side BC = 8 cm, AB = 12 cm, what is the length of side DE and DF?
7. In two congruent quadrilaterals, one diagonal measures 13 cm and the other 17 cm. What are the diagonal lengths in the second quadrilateral?
8. Two congruent trapeziums have angles of 75° and 105°. What are the angles in the other trapezium?
9. If two congruent pentagons have side lengths of 9 cm, what are the corresponding side lengths in the other pentagon?
10. Two congruent parallelograms have angles of 85° and 95°. What are the angles in the other parallelogram?
11. Triangle ABC is congruent to triangle XYZ, and angle ∠A = 55°. What are the measures of angles ∠B and ∠C?
12. If two congruent trapeziums have one base measuring 17 cm and the other 21 cm, what are the base lengths in the second trapezium?
13. In two congruent rhombuses, one diagonal measures 18 cm and the other 22 cm. What are the lengths of the diagonals in the second rhombus?
14. Triangle DEF is congruent to triangle GHI. If side DE = 11 cm and DF = 13 cm, what are the side lengths of sides GH and GI?
15. Two congruent octagons have interior angles of 135°. What are the angles in the other octagon?
16. If triangle PQR is congruent to triangle STU and side QR =

10 cm, PR = 12 cm, what is the length of side TU?

17. In two congruent parallelograms, one diagonal measures 15 cm, and the other diagonal measures 20 cm. What are the diagonal lengths in the second parallelogram?
18. Triangle ABC is congruent to triangle DEF. If angle ∠A = 60°, and side AB = 10 cm, what is the measure of angle ∠D, and the length of side DE?
19. If two congruent hexagons have a side length of 12 cm, what are the corresponding side lengths in the other hexagon?
20. In two congruent quadrilaterals, one angle measures 85°, while the adjacent angle measures 95°. What are the corresponding angles in the second quadrilateral?

Answers with Explanation

Easy Level Answers

1. Answer: $$DE = 6 \, \text{cm}$$
   Explanation: Corresponding sides are equal in congruent figures.
2. Answer: $$8 \, \text{cm}$$
   Explanation: All sides are equal in congruent squares.
3. Answer: $$∠P = 40^\circ$$
   Explanation: Corresponding angles are equal in congruent triangles.
4. Answer: $$12 \, \text{cm}$$
   Explanation: Corresponding sides in congruent rectangles are equal.
5. Answer: $$50^\circ, 60^\circ, 70^\circ$$
   Explanation: Congruent triangles have equal corresponding angles.
6. Answer: $$XZ = 7 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
7. Answer: $$∠A = 90^\circ$$
   Explanation: Corresponding angles in congruent quadrilaterals are equal.
8. Answer: $$5 \, \text{cm}$$
   Explanation: Corresponding sides in congruent pentagons are equal.
9. Answer: $$∠G = 80^\circ$$
   Explanation: Corresponding angles in congruent triangles are equal.
10. Answer: $$10 \, \text{cm} \, \text{and} \, 14 \, \text{cm}$$
    Explanation: Corresponding sides in congruent parallelograms are equal.
11. Answer: $$∠Y = 45^\circ$$
    Explanation: Corresponding angles in congruent triangles are equal.
12. Answer: $$9 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.
13. Answer: $$108^\circ$$
    Explanation: Corresponding angles in congruent pentagons are equal.
14. Answer: $$7 \, \text{cm}$$
    Explanation: Corresponding sides in congruent rhombuses are equal.
15. Answer: $$4 \, \text{cm}$$
    Explanation: Corresponding sides in congruent hexagons are equal.
16. Answer: $$8 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.
17. Answer: $$15 \, \text{cm}$$
    Explanation: Corresponding sides in congruent trapeziums are equal.
18. Answer: $$∠U = 30^\circ$$
    Explanation: Corresponding angles in congruent triangles are equal.
19. Answer: $$135^\circ$$
    Explanation: Corresponding angles in congruent octagons are equal.
20. Answer: $$10 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.

Medium Level Answers

1. Answer: $$∠C = 60^\circ$$
   Explanation: The sum of angles in a triangle is 180°, and the other two angles are given as 50° and 70°. So, $$180^\circ – (50^\circ + 70^\circ) = 60^\circ$$ .
2. Answer: $$60^\circ \, \text{and} \, 120^\circ$$
   Explanation: Corresponding angles in congruent parallelograms are equal.
3. Answer: $$SU = 15 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
4. Answer: $$18 \, \text{cm} \, \text{and} \, 12 \, \text{cm}$$
   Explanation: Corresponding sides in congruent trapeziums are equal.
5. Answer: $$DF = 11 \, \text{cm}, DE = 9 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
6. Answer: $$90^\circ, 45^\circ, 135^\circ, 90^\circ$$
   Explanation: Corresponding angles in congruent quadrilaterals are equal.
7. Answer: $$PQ = 14 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
8. Answer: $$10 \, \text{cm}$$
   Explanation: Corresponding diagonals in congruent rhombuses are equal.
9. Answer: $$7 \, \text{cm}$$
   Explanation: Corresponding sides in congruent pentagons are equal.
10. Answer: $$∠C = 80^\circ$$
    Explanation: The sum of the angles in a triangle is 180°, so $$180^\circ – (45^\circ + 55^\circ) = 80^\circ$$ .
11. Answer: $$20 \, \text{cm}$$
    Explanation: Corresponding sides in congruent trapeziums are equal.
12. Answer: $$∠X = 35^\circ$$
    Explanation: Corresponding angles in congruent triangles are equal.
13. Answer: $$8 \, \text{cm}$$
    Explanation: Corresponding sides in congruent pentagons are equal.
14. Answer: $$12 \, \text{cm} \, \text{and} \, 16 \, \text{cm}$$
    Explanation: Corresponding diagonals in congruent rhombuses are equal.
15. Answer: $$5 \, \text{cm}, 6 \, \text{cm}, 7 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.
16. Answer: $$60^\circ \, \text{and} \, 120^\circ$$
    Explanation: Corresponding angles in congruent trapeziums are equal.
17. Answer: $$∠C = 90^\circ$$
    Explanation: The sum of angles in a triangle is 180°, and $$180^\circ – (50^\circ + 40^\circ) = 90^\circ$$ .
18. Answer: $$10 \, \text{cm}$$
    Explanation: Corresponding sides in congruent hexagons are equal.
19. Answer: $$9 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.
20. Answer: $$15 \, \text{cm}$$
    Explanation: Corresponding sides in congruent squares are equal.

Hard Level Answers

1. Answer: $$70^\circ, 110^\circ, 70^\circ, 110^\circ$$
   Explanation: Corresponding angles in congruent trapeziums are equal.
2. Answer: $$40^\circ, 70^\circ, 70^\circ$$
   Explanation: Corresponding angles in congruent triangles are equal.
3. Answer: $$DE = 12 \, \text{cm}, DF = 15 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
4. Answer: $$108^\circ$$
   Explanation: Corresponding angles in congruent pentagons are equal.
5. Answer: $$PQ = 14 \, \text{cm}, PR = 9 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
6. Answer: $$DE = 8 \, \text{cm}, DF = 12 \, \text{cm}$$
   Explanation: Corresponding sides in congruent triangles are equal.
7. Answer: $$13 \, \text{cm} \, \text{and} \, 17 \, \text{cm}$$
   Explanation: Corresponding diagonals in congruent quadrilaterals are equal.
8. Answer: $$75^\circ \, \text{and} \, 105^\circ$$
   Explanation: Corresponding angles in congruent trapeziums are equal.
9. Answer: $$9 \, \text{cm}$$
   Explanation: Corresponding sides in congruent pentagons are equal.
10. Answer: $$85^\circ \, \text{and} \, 95^\circ$$
    Explanation: Corresponding angles in congruent parallelograms are equal.
11. Answer: $$∠B = 65^\circ, ∠C = 60^\circ$$
    Explanation: The sum of angles in a triangle is 180°, and $$180^\circ – 55^\circ = 125^\circ \text{ for ∠B and ∠C combined.}$$
12. Answer: $$17 \, \text{cm} \, \text{and} \, 21 \, \text{cm}$$
    Explanation: Corresponding sides in congruent trapeziums are equal.
13. Answer: $$18 \, \text{cm} \, \text{and} \, 22 \, \text{cm}$$
    Explanation: Corresponding diagonals in congruent rhombuses are equal.
14. **Answer

:** $$GH = 11 \, \text{cm}, GI = 13 \, \text{cm}$$
Explanation: Corresponding sides in congruent triangles are equal.

15. Answer: $$135^\circ$$
    Explanation: Corresponding angles in congruent octagons are equal.
16. Answer: $$QR = 10 \, \text{cm}, PR = 12 \, \text{cm}$$
    Explanation: Corresponding sides in congruent triangles are equal.
17. Answer: $$15 \, \text{cm} \, \text{and} \, 20 \, \text{cm}$$
    Explanation: Corresponding diagonals in congruent parallelograms are equal.
18. Answer: $$∠D = 60^\circ, DE = 10 \, \text{cm}$$
    Explanation: Corresponding angles and sides in congruent triangles are equal.
19. Answer: $$12 \, \text{cm}$$
    Explanation: Corresponding sides in congruent hexagons are equal.
20. Answer: $$85^\circ, 95^\circ$$
    Explanation: Corresponding angles in congruent quadrilaterals are equal.