Congruent figures
Two coins of the same denomination, or two pages cut from one stencil, match exactly when one is placed over the other.
Two figures are congruent (≅) if they have the same shape and same size, so one fits exactly over the other (possibly after sliding, turning or flipping). Congruent figures have equal areas and equal perimeters.
In the figure: Two congruent triangles: matching marks show equal sides.
Rule 1: Test by placing over — If every part matches exactly after slide, turn or flip, the figures are congruent.
- Easy: Two circles of radius 5 cm are congruent.
- Medium: Squares of side 4 cm and 5 cm are not congruent.
- Exam-type: A 4 × 3 rectangle turned sideways is a 3 × 4 rectangle: congruent.
Rule 2: What congruent figures share — Equal area, equal perimeter, equal matching sides and angles.
- Easy: If one has area 24, so does the other.
- Medium: If one perimeter is 18 cm, so is the other.
- Exam-type: Equal area alone does not make shapes congruent: 6 × 2 and 4 × 3 both have area 12.
Rule 3: Similar is not congruent — Same shape but different size means not congruent (an enlargement).
- Easy: A photo and its enlargement are not congruent.
- Medium: Triangle 3, 4, 5 and triangle 6, 8, 10 are not congruent.
- Exam-type: Two right-angled triangles with legs 3, 4 are congruent.
Congruent = twins: same shape and same size.
Calling two figures congruent because their areas match. Shape must match too.
Trace one figure on tracing paper and slide it over the other. Memory clue: "Twins fit on twins."
Spot the difference
| Term | Shape | Size |
|---|---|---|
| Congruent | same | same |
| Similar | same | different |
Practice questions
- 1Are two circles of radius 5 cm congruent?
- 2Are squares of side 4 cm and 5 cm congruent?
- 3Two congruent figures: one has area 24. Area of the other?
- 4A 4 × 3 rectangle is turned sideways. Is it congruent to the original?
- 5Rectangles 6 × 2 and 4 × 3 have the same area. Congruent?
- 6Perimeter of one of two congruent figures is 18 cm. The other?
- 7Triangles 3, 4, 5 and 6, 8, 10: congruent?
- 8Two congruent squares have perimeter 20 cm. Side?
Show answers (try them yourself first)
- 1Yes
- 2No
- 324
- 4Yes
- 5No
- 618 cm
- 7No (one is an enlargement)
- 85 cm
Congruence of triangles: SSS
A triangular roof frame built with beams of 3, 4, 5 m has only one possible shape, so any two such frames are identical.
If the three sides of one triangle equal the three sides of another, the triangles are congruent. This is the SSS (side-side-side) rule. Triangles are rigid: three sides fix the shape.
In the figure: Matching marks (1, 2, 3 ticks) show the three pairs of equal sides.
Rule 1: Check three pairs — Arrange the sides of both triangles in order and compare.
- Easy: 3, 4, 5 and 5, 3, 4 → same sides → congruent.
- Medium: 6, 7, 8 and 6, 7, 9 → not congruent.
- Exam-type: 5, 5, 8 and 8, 5, 5 → congruent.
Rule 2: Name the matching — Write △ABC ≅ △PQR so that A↔P, B↔Q, C↔R.
- Easy: AB = PQ, BC = QR, CA = RP.
- Medium: Side opposite A matches side opposite P.
- Exam-type: Order of letters matters.
Rule 3: Why equal angles alone fail — Three equal angles (AAA) give the same shape but not necessarily the same size.
- Easy: Equilateral triangles of side 3 and 6 both have 60°.
- Medium: They are not congruent.
- Exam-type: AAA is not a congruence rule.
SSS needs all three pairs of sides equal.
Matching sides in the wrong order, such as 3 with 5. Sort both lists first.
Sort sides from small to large before comparing. Memory clue: "Three sides, three pairs, same pairs."
Spot the difference
| Rule | Needs | Enough? |
|---|---|---|
| SSS | 3 sides | Yes |
| AAA | 3 angles | No |
Practice questions
- 1Triangle 3, 4, 5 and triangle 5, 3, 4: congruent?
- 2Triangle 6, 7, 8 and 6, 7, 9: congruent?
- 3Which rule uses three sides?
- 4Triangle 5, 5, 8 and 8, 5, 5: congruent?
- 5△ABC ≅ △PQR with AB = 5. Which side equals 5?
- 6Are two equilateral triangles with all angles 60° always congruent?
- 7Triangles 7, 9, 12 and 9, 12, 7: congruent?
- 8△XYZ ≅ △LMN. Which vertex matches Y?
Show answers (try them yourself first)
- 1Yes (SSS)
- 2No
- 3SSS
- 4Yes (SSS)
- 5PQ
- 6No: sizes can differ
- 7Yes (SSS)
- 8M
Congruence criteria: SAS, ASA and RHS
A tailor cuts two identical triangular patches using only two sides and the angle between them as the measurements.
Triangles are congruent if two sides and the included angle (SAS), or two angles and the included side (ASA), or in right triangles the hypotenuse and one side (RHS) are equal. SSA (angle not between the sides) and AAA do not work.
In the figure: SSS, SAS, ASA and RHS: four sets of facts that fix a triangle.
Congruence criteria: SAS, ASA and RHS
In the app you can move this figure with your finger.
Open in the appRule 1: SAS — Two sides and the angle between them.
- Easy: AB = PQ, ∠B = ∠Q, BC = QR → SAS.
- Medium: Sides 5, 6 with 50° between in both → congruent.
- Exam-type: Angle not between the sides (SSA) is not enough.
Rule 2: ASA — Two angles and the side between them (the third angle then follows).
- Easy: ∠A = ∠P, AB = PQ, ∠B = ∠Q → ASA.
- Medium: Angles 40°, 60° with side 7 between → fixed triangle.
- Exam-type: Two angles and a side not between them (AAS) also works, since the third angle is known.
Rule 3: RHS (right triangles) — Right angle, equal hypotenuse and one equal side.
- Easy: Hypotenuse 13 and one side 5 in both → congruent.
- Medium: The other side is then 12 in both.
- Exam-type: Works only when a right angle is present.
The angle in SAS is between the two sides; the side in ASA is between the two angles.
Using SSA (two sides and a non-included angle). It can give two different triangles.
Mark the equal parts with ticks and arcs on the figure. Memory clue: "SAS sandwich, ASA sandwich, RHS right."
Spot the difference
| Rule | Given | Valid? |
|---|---|---|
| SAS | side, angle between, side | Yes |
| ASA | angle, side between, angle | Yes |
| SSA | two sides, other angle | No |
Practice questions
- 1AB = PQ, ∠B = ∠Q, BC = QR. Which rule?
- 2∠A = ∠P, AB = PQ, ∠B = ∠Q. Which rule?
- 3Is AAA a congruence rule?
- 4Right triangles with hypotenuse 13 and a side 5 in both: congruent? Other side?
- 5Two sides 5, 6 and angle 50° between them in both: congruent?
- 6Two sides and a non-included angle: is it enough?
- 7In ASA, angles are 40° and 60°. Find the third.
- 8Angles 40°, 60° and a side of 7 not between them: enough?
Show answers (try them yourself first)
- 1SAS
- 2ASA
- 3No
- 4Yes (RHS); 12
- 5Yes (SAS)
- 6No (SSA)
- 780°
- 8Yes (AAS)
Corresponding parts of congruent triangles
Cut a paper triangle and trace it twice. Every side and angle of the first has a partner of equal size in the second.
When △ABC ≅ △PQR, the corresponding parts are equal: AB = PQ, BC = QR, CA = RP and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R. The pairing follows the order of the letters.
In the figure: A ↔ P, B ↔ Q, C ↔ R: each part has a matching partner.
Rule 1: Read the matching from letters — The first letters match, the second letters match, the third letters match.
- Easy: △ABC ≅ △PQR: AB ↔ PQ.
- Medium: △DEF ≅ △XYZ: ∠E ↔ ∠Y.
- Exam-type: △ABC ≅ △QPR: AB ↔ QP, ∠C ↔ ∠R.
Rule 2: Use equal parts to find unknowns — Copy the known value across to the matching part.
- Easy: △ABC ≅ △PQR, AB = 7 → PQ = 7.
- Medium: ∠C = 65° → ∠R = 65°.
- Exam-type: Perimeter of △PQR equals perimeter of △ABC.
Rule 3: Prove then conclude — First show the triangles congruent by a rule; then the other parts are equal.
- Easy: AB = PQ, BC = QR, CA = RP → congruent by SSS.
- Medium: So ∠A = ∠P (corresponding angles).
- Exam-type: A common side counts as an equal pair.
The order of letters in △ABC ≅ △PQR gives the matching.
Matching AB with QR. Match first with first, second with second.
Underline the letters in both names and read them in pairs. Memory clue: "First with first, second with second."
Spot the difference
| In △ABC ≅ △PQR | Side | Angle |
|---|---|---|
| 1st pair | AB = PQ | ∠A = ∠P |
| 2nd pair | BC = QR | ∠B = ∠Q |
| 3rd pair | CA = RP | ∠C = ∠R |
Practice questions
- 1If △ABC ≅ △PQR, which side matches AB?
- 2△ABC ≅ △PQR, AB = 7. Find PQ.
- 3△ABC ≅ △PQR, ∠C = 65°. Find ∠R.
- 4△DEF ≅ △XYZ. Which angle matches ∠E?
- 5△ABC ≅ △QPR. Which side matches AB?
- 6Perimeter of △ABC is 21 cm. Perimeter of △PQR (congruent)?
- 7△ABC ≅ △PQR with AB = 5, BC = 6, CA = 7. Find PQ + QR + RP.
- 8△ABC ≅ △PQR, ∠A = 50°, ∠B = 60°. Find ∠R.
Show answers (try them yourself first)
- 1PQ
- 27
- 365°
- 4∠Y
- 5QP
- 621 cm
- 718
- 870°
Angles of isosceles and equilateral triangles
A tent with equal slanting sides leans the same way on both sides, so its two bottom corners are equal.
In an isosceles triangle the angles opposite the equal sides (the base angles) are equal. In an equilateral triangle all three angles are 60°. This follows by splitting the triangle into two congruent halves.
In the figure: Apex 40°, so each base angle is (180° – 40°) ÷ 2 = 70°.
Rule 1: Isosceles: find base angles — Base angle = (180° – apex) ÷ 2.
- Easy: Apex 40° → base angles 70° each.
- Medium: Apex 100° → base angles 40° each.
- Exam-type: Apex 20° → base angles 80° each.
Rule 2: Isosceles: find the apex — Apex = 180° – 2 × base angle.
- Easy: Base angle 50° → apex 80°.
- Medium: Base angle 65° → apex 50°.
- Exam-type: Right isosceles triangle: base angles 45° each.
Rule 3: Equilateral — All angles 60°, since 180° ÷ 3 = 60°.
- Easy: Each angle is 60°.
- Medium: A triangle with angles 60°, 60°, x has x = 60°.
- Exam-type: An equilateral triangle is also isosceles.
Equal sides have equal angles opposite them.
Using 180° ÷ 2 for the base angles. First subtract the apex, then halve.
Subtract the odd angle from 180°, then halve. Memory clue: "Equal sides, equal angles; subtract, then split."
Spot the difference
| Triangle | Angles | Example |
|---|---|---|
| Equilateral | all equal, 60° | 60, 60, 60 |
| Isosceles | two equal | 70, 70, 40 |
| Scalene | none equal | 30, 60, 90 |
Practice questions
- 1Isosceles triangle, apex 40°: find a base angle.
- 2Isosceles triangle, base angle 50°: find the apex.
- 3What is each angle of an equilateral triangle?
- 4Isosceles, apex 100°: base angle?
- 5Isosceles, base angle 65°: apex?
- 6Right isosceles triangle: base angle?
- 7In an isosceles triangle the apex is twice a base angle. Find all angles.
- 8Isosceles, base angle (2x + 10)° and apex 40°: find x.
Show answers (try them yourself first)
- 170°
- 280°
- 360°
- 440°
- 550°
- 645°
- 745°, 45°, 90°
- 830