Introduction
Congruent figures are shapes that are identical in size and shape, though their orientation or position might differ. When two figures are congruent, all corresponding angles and sides are equal. This concept is fundamental in geometry and is often tested in Key Stage 3 exams to assess students’ understanding of shape transformations and spatial reasoning.
Key Points to Remember
- Congruent figures have the same size and shape.
- Corresponding sides of congruent figures are equal in length.
- Corresponding angles of congruent figures are equal in measure.
- Congruent figures can be rotated, flipped (reflected), or translated (moved) and still remain congruent.
To identify congruent figures, you can:
- Compare their side lengths.
- Check their angles.
- See if one shape can be transformed into another through reflection, rotation, or translation.
Question Set on Identifying Congruent Figures
Easy Level
Identify Congruent Figures
Q1: Are two triangles with sides of 3 cm, 4 cm, and 5 cm congruent to each other?
Q2: Can two squares with side lengths of 5 cm be congruent?
Q3: If two circles have the same radius, are they congruent?
Q4: Are two rectangles with dimensions of 6 cm by 8 cm and 8 cm by 6 cm congruent?
Q5: Identify if two isosceles triangles with equal sides of 6 cm and 6 cm, and a base of 4 cm, are congruent.
Q6: Can two parallelograms with side lengths of 7 cm and 3 cm be congruent?
Q7: Are two hexagons with side lengths of 4 cm congruent?
Q8: If two rhombuses have all sides equal to 5 cm, are they congruent?
Q9: Are two trapeziums congruent if their non-parallel sides measure the same?
Q10: Are two equilateral triangles with side lengths of 8 cm congruent?
Transformation and Congruence
Q11: Can a triangle be congruent to itself after a rotation of 90 degrees?
Q12: If you reflect a square over a line, is it congruent to its original shape?
Q13: Are two rectangles congruent if they are reflections of each other?
Q14: Can two congruent triangles be oriented differently and still remain congruent?
Q15: Are two circles congruent if one is rotated by 180 degrees?
Q16: Identify if two trapezoids with equal base lengths and heights are congruent after a translation.
Q17: Can two identical pentagons be congruent if one is reflected?
Q18: If a figure is translated 3 units to the left, is it congruent to its original position?
Q19: Can two congruent parallelograms have different orientations?
Q20: Are two squares congruent if one is rotated by 45 degrees?
Medium Level
Prove Congruence
Q1: Given two triangles, one with sides 6 cm, 8 cm, and 10 cm, and the other with sides 8 cm, 6 cm, and 10 cm, are they congruent? Prove it.
Q2: Are two rectangles with dimensions 7 cm by 5 cm and 5 cm by 7 cm congruent? Justify your answer.
Q3: Prove that two right-angled triangles with leg lengths of 9 cm and 12 cm are congruent.
Q4: If two parallelograms have sides 8 cm and 10 cm, but their angles are different, are they congruent?
Q5: Two quadrilaterals have all sides equal but different angles. Are they congruent?
Q6: Can two congruent triangles have different perimeters?
Q7: Are two circles with equal diameters congruent?
Q8: If a rhombus is reflected over a line of symmetry, is it congruent to the original?
Q9: Prove that two trapezoids with identical base lengths and non-parallel sides are congruent.
Q10: Can two rectangles with the same diagonal length but different side lengths be congruent?
Identifying Transformations
Q11: A triangle is rotated by 90 degrees. Is the new figure congruent to the original triangle?
Q12: Two congruent triangles are placed such that their corresponding sides are parallel. Are they still congruent?
Q13: After a reflection, are two parallelograms congruent?
Q14: If a shape is translated 5 units down and 3 units to the right, is it still congruent to its original?
Q15: Identify whether two equilateral triangles are congruent after one is rotated by 120 degrees.
Q16: Can two quadrilaterals with equal side lengths but different orientations be congruent?
Q17: Prove that two congruent rectangles remain congruent after a rotation of 90 degrees.
Q18: Are two hexagons congruent if one is a translation of the other?
Q19: Can two congruent trapezoids have different areas?
Q20: Are two congruent shapes necessarily in the same orientation?
Hard Level
Advanced Proofs and Applications
Q1: Given two scalene triangles with corresponding sides proportional but not equal, are they congruent? Explain why or why not.
Q2: Two rectangles have equal areas but different dimensions. Are they congruent?
Q3: Prove that two right-angled triangles with equal hypotenuses and one equal leg are congruent.
Q4: A rectangle is reflected over a diagonal. Is the resulting figure congruent to the original?
Q5: Prove that two rhombuses with equal side lengths and one equal angle are congruent.
Q6: Two parallelograms have equal diagonals but different angles. Are they congruent?
Q7: Prove that two congruent hexagons remain congruent after one is rotated by 60 degrees.
Q8: Are two trapezoids with identical side lengths but different angles congruent? Justify your answer.
Q9: Prove that two equilateral triangles remain congruent after a translation followed by a reflection.
Q10: If two quadrilaterals have the same perimeter but different side lengths, are they congruent?
Transformations and Proving Congruence
Q11: A triangle is reflected over a vertical line, then rotated by 180 degrees. Is the new figure congruent to the original?
Q12: Two quadrilaterals with corresponding sides equal but angles differing by 10 degrees are compared. Are they congruent?
Q13: Prove that a square remains congruent to itself after a reflection followed by a rotation.
Q14: Identify if two congruent pentagons remain congruent after one undergoes a series of transformations, including translation and reflection.
Q15: Prove that two rectangles with equal perimeters but different side lengths are not congruent.
Q16: Given two rhombuses with equal side lengths but different internal angles, are they congruent?
Q17: Are two congruent hexagons still congruent if one is translated 5 units up and then reflected?
Q18: Prove that two congruent right-angled triangles remain congruent after one is reflected over a horizontal axis.
Q19: Identify if two congruent shapes have equal areas regardless of their orientation.
Q20: Prove that two rectangles with the same diagonal length are congruent only if their sides are also equal.
Easy Level Answers and Explanations
Q1: Yes.
Explanation: Both triangles have sides of 3 cm, 4 cm, and 5 cm, so they are congruent by the Side-Side-Side (SSS) rule.
Q2: Yes.
Explanation: All squares with equal side lengths are congruent because their sides and angles are identical.
Q3: Yes.
Explanation: Circles with the same radius are congruent, as their size and shape are identical.
Q4: Yes.
Explanation: Rectangles with the same side lengths, regardless of their orientation, are congruent.
Q5: Yes.
Explanation: Two isosceles triangles with equal sides and bases are congruent by the SSS rule.
Q6: Yes.
Explanation: Parallelograms with identical side lengths are congruent, provided their angles are the same.
Q7: Yes.
Explanation: Regular hexagons with equal side lengths are congruent because their angles and sides are identical.
Q8: Yes.
Explanation: Rhombuses with equal side lengths are congruent, as all sides and angles match.
Q9: No.
Explanation: For trapeziums to be congruent, their non-parallel sides and angles must also be the same.
Q10: Yes.
Explanation: Equilateral triangles with the same side lengths are always congruent.
Q11: Yes.
Explanation: A triangle is congruent to itself even after a rotation, as its sides and angles remain unchanged.
Q12: Yes.
Explanation: A square reflected over a line is congruent to its original because its side lengths and angles remain unchanged.
Q13: Yes.
Explanation: Reflections do not change the dimensions or angles of rectangles, so they remain congruent.
Q14: Yes.
Explanation: Congruence is unaffected by the orientation of the shape, so triangles can have different orientations and still be congruent.
Q15: Yes.
Explanation: Rotating a circle does not affect its size or shape, so it remains congruent to its original.
Q16: Yes.
Explanation: Translation does not change the size or shape of a trapezoid, so it remains congruent.
Q17: Yes.
Explanation: Reflected shapes remain congruent because reflections do not alter side lengths or angles.
Q18: Yes.
Explanation: Translations preserve the size and shape of a figure, so the figure remains congruent to its original.
Q19: Yes.
Explanation: Parallelograms can be oriented differently, but as long as their side lengths and angles are the same, they remain congruent.
Q20: Yes.
Explanation: Rotation does not affect the side lengths or angles of a square, so it remains congruent.
Medium Level Answers and Explanations
Q1: Yes.
Explanation: The triangles have the same side lengths, though in a different order. By the SSS rule, they are congruent.
Q2: Yes.
Explanation: Rectangles with the same side lengths, even if the dimensions are presented in a different order, are congruent.
Q3: Yes.
Explanation: Right-angled triangles with the same leg lengths are congruent by the Pythagorean Theorem and the SSS rule.
Q4: No.
Explanation: While the side lengths are the same, the difference in angles means the parallelograms are not congruent.
Q5: No.
Explanation: For two quadrilaterals to be congruent, both their side lengths and angles must be the same.
Q6: No.
Explanation: Congruent triangles must have equal perimeters, as their corresponding sides must be equal in length.
Q7: Yes.
Explanation: Circles with the same diameter (or radius) are always congruent.
Q8: Yes.
Explanation: Reflection over a line does not change the side lengths or angles of a rhombus, so it remains congruent.
Q9: Yes.
Explanation: Trapezoids with identical base lengths and non-parallel sides are congruent if all other corresponding measurements match.
Q10: No.
Explanation: The rectangles have the same diagonal, but if their side lengths differ, they are not congruent.
Q11: Yes.
Explanation: Rotating a triangle does not alter its side lengths or angles, so it remains congruent.
Q12: Yes.
Explanation: If the corresponding sides of the triangles are parallel and of equal length, the triangles remain congruent.
Q13: Yes.
Explanation: Reflections do not affect the size or shape of parallelograms, so they remain congruent.
Q14: Yes.
Explanation: Translations preserve congruence, as the size and shape of the original figure remain unchanged.
Q15: Yes.
Explanation: Equilateral triangles with the same side lengths remain congruent even after rotation.
Q16: Yes.
Explanation: Orientation does not affect congruence, so quadrilaterals with equal side lengths remain congruent.
Q17: Yes.
Explanation: Rotating a rectangle does not change its side lengths or angles, so it remains congruent.
Q18: Yes.
Explanation: Hexagons remain congruent after a translation, as their side lengths and angles are preserved.
Q19: No.
Explanation: Two trapezoids can have the same side lengths but different areas if their angles are different.
Q20: No.
Explanation: Two congruent shapes can have different orientations but will still have the same size and shape.
Hard Level Answers and Explanations
Q1: No.
Explanation: For two triangles to be congruent, their corresponding sides must be equal, not just proportional.
Q2: No.
Explanation: Congruence requires equal side lengths, not just equal areas.
Q3: Yes.
Explanation: Right-angled triangles with the same hypotenuse and one equal leg are congruent by the Hypotenuse-Leg (HL) rule.
Q4: Yes.
Explanation: Reflection over a diagonal does not alter the dimensions of a rectangle, so it remains congruent.
Q5: Yes.
Explanation: Rhombuses with equal side lengths and at least one equal angle are congruent, as their other angles and side lengths will also match.
Q6: No.
Explanation: Parallelograms with different angles cannot be congruent, even if their diagonals are the same.
Q7: Yes.
Explanation: Regular hexagons remain congruent after any rotation, as all sides and angles are preserved.
Q8: No.
Explanation: For trapezoids to be congruent, their corresponding angles must also be equal, not just their side lengths.
Q9: Yes.
Explanation: A translation followed by a reflection preserves the size and shape of a figure, so the triangles remain congruent.
Q10: No.
Explanation: Having the same perimeter does not imply congruence if the side lengths are different.
Q11: Yes.
Explanation: A reflection followed by a rotation preserves the dimensions of a figure, so it remains congruent.
Q12: No.
Explanation: Congruent quadrilaterals must have equal angles in addition to equal side lengths, so a difference in angles means they are not congruent.
Q13: Yes.
Explanation: A square remains congruent after both reflection and rotation, as its sides and angles are unchanged.
Q14: Yes.
Explanation: Transformations like translation and reflection do not affect congruence, so the pentagons remain congruent.
Q15: No.
Explanation: Rectangles with equal perimeters but different side lengths are not congruent, as congruence requires equal side lengths.
Q16: No.
Explanation: Rhombuses must have equal angles as well as equal side lengths to be congruent.
Q17: Yes.
Explanation: A translation followed by a reflection preserves the size and shape of hexagons, so they remain congruent.
Q18: Yes.
Explanation: Right-angled triangles remain congruent after reflection, as the side lengths and angles are preserved.
Q19: Yes.
Explanation: Congruent shapes always have the same area, regardless of their orientation.
Q20: Yes, provided their sides are equal.
Explanation: Rectangles with the same diagonal are congruent only if their corresponding side lengths are equal.
