K2KARAKORAM
मराठीEnglish
Open the app

State Board · Class 8 · Mathematics

Lesson 13: Congruence of Triangles

4 concepts · 32 practice questions with answers · Free to read

This lesson is also in: मराठी

Congruent triangles and correspondence

From daily life

Two photocopies of the same note fit exactly on top of each other. Such shapes are congruent.

Simple definition

Two triangles are congruent (≅) if one fits exactly on the other. Their three sides and three angles are equal one by one.

ABCPQR≅

In the figure: ΔABC ≅ ΔPQR: sides with the same tick marks are equal (AB = PQ, AC = PR, BC = QR) and ∠B = ∠Q.

Formula: ΔABC ≅ ΔPQR means A↔P , B↔Q , C↔R

In words: The order of letters matters: first with first, second with second, third with third.

Live figure

Congruent triangles and correspondence

In the app you can move this figure with your finger.

Open in the app
  • Easy: ΔABC ≅ ΔPQR. AB = 5 cm so PQ = 5 cm. ∠B = 60° so ∠Q = 60°. The partner of side AC is PR.
  • Medium: ΔABC ≅ ΔXYZ, ∠A = 40°, ∠B = 75°: ∠X = 40°, ∠Y = 75°; ∠Z = ∠C = 180 − 40 − 75 = 65°.
  • Exam-type: ΔDEF ≅ ΔKLM, DE = 7, EF = 9, FD = 11: KL = 7, LM = 9, MK = 11; the perimeter of both is 7 + 9 + 11 = 27.
Remember

Congruent triangles also have equal area and perimeter.

Common mistake

If you write ΔABC ≅ ΔQPR then A↔Q and B↔P! Changing the order changes the pairs.

Shortcut

Write the pairs straight from the order of letters: ΔABC ≅ ΔPQR ⇒ A↔P, B↔Q, C↔R. Memory clue: Pairs follow the order of the letters: first-first, second-second, third-third.

Spot the difference

PointCongruent trianglesOnly angles equal
Sidesall three pairs equalcan be bigger or smaller
Anglesall three equalall three equal
Sizeexactly the samesame shape but bigger or smaller
Example3, 4, 5 and 3, 4, 53, 4, 5 and 6, 8, 10

Practice questions

Easy
  1. 1ΔABC ≅ ΔPQR and AB = 7 cm. PQ = ?
  2. 2ΔABC ≅ ΔPQR and ∠A = 55°. ∠P = ?
  3. 3If ΔABC ≅ ΔXYZ, which side corresponds to BC?
Medium
  1. 4ΔLMN ≅ ΔDEF, ∠M = 80°, ∠N = 45°. ∠D = ?
  2. 5ΔABC ≅ ΔPQR, AB = 6, BC = 8, CA = 10. Perimeter of ΔPQR?
  3. 6If ΔABC ≅ ΔQRP, which side corresponds to AB and which angle to ∠B?
Challenging
  1. 7ΔABC ≅ ΔPQR. The area of ΔABC is 30 sq cm and its perimeter 30 cm. Area and perimeter of ΔPQR?
  2. 8ΔABC ≅ ΔDEF, AB = 3x + 1 and DE = 16. x = ?
Show answers (try them yourself first)
  1. 1PQ = AB = 7 cm.
  2. 2∠P = ∠A = 55°.
  3. 3B↔Y, C↔Z so BC ↔ YZ.
  4. 4∠L = 180 − 80 − 45 = 55°; ∠D = ∠L = 55°.
  5. 56 + 8 + 10 = 24 (congruent ⇒ equal perimeters).
  6. 6A↔Q, B↔R, C↔P; AB ↔ QR; ∠B ↔ ∠R.
  7. 7Both are the same: area 30 sq cm, perimeter 30 cm.
  8. 8AB = DE; 3x + 1 = 16; 3x = 15; x = 5.

Tests of congruence: SSS, SAS, ASA, AAS

From daily life

A carpenter does not measure everything to check chair legs; a few key measures are enough. For triangles too we need not check all six measures.

Simple definition

If three suitable measures match, two triangles are congruent. Here are the four tests.

SSSSASASAAAS

In the figure: Marks of the four tests: the same marks mean equal sides or equal angles. In SAS the angle is between the two sides; in ASA the side is between the two angles.

The four tests

TestWhat must be equalExample
SSSthree sides4, 5, 6 and 4, 5, 6
SAStwo sides and the angle between them7 cm, 40°, 5 cm
ASAtwo angles and the side between them50°, 6 cm, 70°
AAStwo angles and a side that is not between them60°, 70° and 8 cm (not between)
Live figure

Tests of congruence: SSS, SAS, ASA, AAS

In the app you can move this figure with your finger.

Open in the app
  • Easy: ΔABC: AB = 4, BC = 5, CA = 6 ; ΔXYZ: XY = 4, YZ = 5, ZX = 6. Three sides equal → by SSS, ΔABC ≅ ΔXYZ.
  • Medium: ΔPQR: PQ = 7, ∠Q = 40°, QR = 5 ; ΔLMN: LM = 7, ∠M = 40°, MN = 5. The angle is between the two sides → by SAS, ΔPQR ≅ ΔLMN.
  • Exam-type: ΔABC: ∠B = 60°, ∠C = 70°, AB = 8 ; ΔDEF: ∠E = 60°, ∠F = 70°, DE = 8. AB is not between ∠B and ∠C, so by AAS ΔABC ≅ ΔDEF. (The third angle is 180 − 60 − 70 = 50° in both.)
Remember

Matching just the right three measures makes two triangles congruent: SSS, SAS, ASA, AAS.

Common mistake

"Two sides and an angle" is not always SAS; the angle must be between the two sides.

Shortcut

Trick for the included angle: it is at the point where the two sides meet. PQ and QR → ∠Q. Memory clue: SSS, SAS, ASA, AAS: three sides; side-angle-side; angle-side-angle; angle-angle-side. AAA and SSA do not work.

Spot the difference

PointSASSSA
Where the angle isbetween the two sidesnot between them
Does it workyesno
Reasononly one triangle formstwo triangles are possible
What is equaltwo sides and the included angletwo sides and a non-included angle

Practice questions

Easy
  1. 1Two triangles have sides 5, 6, 7 and 5, 6, 7. Which test?
  2. 2Both triangles have 4 cm, 60° and 6 cm (the angle in between). Which test?
  3. 3In both, ∠A = 50°, AB = 6 cm, ∠B = 70°. Which test?
Medium
  1. 4In both, ∠P = 45°, ∠Q = 65°, QR = 9. Which test?
  2. 5Both have sides 8, 9 and the 30° angle opposite the side 9. Does a test work?
  3. 6ΔABC: ∠A = 40°, ∠B = 60°; ΔDEF: ∠D = 40°, ∠E = 60°. Can we call them congruent from this alone?
Challenging
  1. 7In ΔPQR and ΔPSR, PQ = PS, QR = SR and PR is common. Which test, and what about ∠Q and ∠S?
  2. 8In ΔABC, AB = AC and D is the midpoint of BC. Why ΔABD ≅ ΔACD and why ∠B = ∠C?
Show answers (try them yourself first)
  1. 1All three sides equal ⇒ SSS.
  2. 2Two sides and the included angle ⇒ SAS.
  3. 3Two angles and the included side ⇒ ASA.
  4. 4QR is not between P and Q ⇒ AAS.
  5. 5The angle is not between the sides ⇒ SSA; this test does not work.
  6. 6No: all three angles (AAA) are equal but the sides can be bigger or smaller.
  7. 7All three sides equal ⇒ SSS; corresponding angles ∠Q = ∠S.
  8. 8AB = AC, BD = DC, AD common ⇒ SSS; corresponding angles ∠B = ∠C.

Hypotenuse-side test (right triangles)

From daily life

Two 13-ft ladders lean on a wall, both with their feet 5 ft from the wall. Both reach 12 ft up the wall: 13² − 5² = 144 and √144 = 12.

Simple definition

Two right triangles with equal hypotenuses and one pair of equal sides are congruent. This is the hypotenuse-side (RHS) test.

5121351213≅

In the figure: Two right triangles: equal hypotenuse 13 (violet) and equal side 5 ⇒ congruent by RHS; the third side is 12 in both.

Formula: right angle + equal hypotenuse + one equal side ⇒ congruent

Live figure

Hypotenuse-side test (right triangles)

In the app you can move this figure with your finger.

Open in the app
  • Easy: ΔABC (∠B = 90°): AC = 13, AB = 5 ; ΔPQR (∠Q = 90°): PR = 13, PQ = 5. By RHS ΔABC ≅ ΔPQR. The third side is 12 in both.
  • Medium: Two right triangles: hypotenuse 10, one side 6. The other side is √(100 − 36) = √64 = 8 in both. So they are congruent by RHS.
  • Exam-type: ΔABC (∠B = 90°): AC = 17, AB = 8 and ΔPQR (∠Q = 90°): PR = 17, PQ = 8 ⇒ congruent by RHS; BC = QR = √(289 − 64) = √225 = 15.
Remember

In RHS three things are equal: the right angle, the hypotenuse and one side.

Common mistake

RHS works only for right triangles. For other triangles "hypotenuse + one side" is not enough.

Shortcut

If the hypotenuse and one side are known, the third side comes from Pythagoras; it is the same in both. Memory clue: Hypotenuse-side works only in a right triangle: "hypotenuse + one side = congruent".

Spot the difference

PointRHSSSS
For which trianglesonly right trianglesany triangle
What must be equalright angle, hypotenuse, one sideall three sides
Third sidefound by Pythagorasmust be checked
Example13 and 5, right angle4, 5, 6

Practice questions

Easy
  1. 1Two right triangles have hypotenuse 13 and one side 5. Which test, and the third side?
  2. 2Hypotenuse 10, one side 8. The other side?
  3. 3Hypotenuse 25, one side 7. The other side?
Medium
  1. 4In RHS which three parts must be equal?
  2. 5In ΔPQR, ∠Q = 90°, PR = 26, PQ = 10. QR = ?
  3. 6One right triangle has hypotenuse 15 and a side 9; another has hypotenuse 15 and a side 12. Are they congruent?
Challenging
  1. 7Two ladders, each 20 ft long, stand 12 ft from a wall. How high on the wall do they reach, and why are the triangles congruent?
  2. 8In rectangle PQRS why is ΔPQR ≅ ΔSRQ, by which test? What follows about the diagonals PR and QS?
Show answers (try them yourself first)
  1. 1RHS; third side = √(169 − 25) = 12.
  2. 2√(100 − 64) = 6.
  3. 3√(625 − 49) = √576 = 24.
  4. 4The right angle, the hypotenuse and one side.
  5. 5√(676 − 100) = √576 = 24.
  6. 6The first has other side √(225 − 81) = 12; the second has other side √(225 − 144) = 9; both have sides 9, 12, 15 ⇒ congruent.
  7. 7Height = √(400 − 144) = √256 = 16 ft; hypotenuse 20 and side 12 are equal in both ⇒ RHS.
  8. 8PQ = SR, ∠PQR = ∠SRQ = 90°, QR common ⇒ SAS; corresponding parts PR = QS (the diagonals are equal).

Using congruence, and tests that do not work

From daily life

Two roads cross at O and O is the midpoint of both. Then the opposite threads joining their ends are equal in length — thanks to congruence.

Simple definition

Corresponding parts of congruent triangles are equal. So once congruence is shown, we can say that sides or angles are equal.

ABCDO

In the figure: AB and CD bisect each other at O: OA = OB, OC = OD and ∠AOC = ∠BOD (vertically opposite) ⇒ ΔAOC ≅ ΔBOD by SAS ⇒ AC = BD.

  • Easy: AB and CD bisect each other at O. OA = OB, OC = OD and ∠AOC = ∠BOD (vertically opposite). By SAS ΔAOC ≅ ΔBOD, so AC = BD.
  • Medium: In ΔABC, AB = AC and AD ⟂ BC. ΔABD and ΔACD are right triangles with AB = AC and AD common. By RHS they are congruent, so BD = DC.
  • Exam-type: Equilateral triangles with sides 3 cm and 6 cm both have 60°, 60°, 60° — equal angles but not congruent.
Live figure

Using congruence, and tests that do not work

In the app you can move this figure with your finger.

Open in the app

In words: Tests that do not work: AAA (three equal angles) — same shape but possibly different size. SSA (two sides and an angle that is not between them) — sometimes two different triangles are possible.

Which tests work

TestWorks?Reason
SSS, SAS, ASA, AASYesonly one triangle is possible
RHSYes (right angle)one right triangle from hypotenuse + side
AAANosize may differ
SSANotwo triangles possible
Remember

Once congruence is proved, the corresponding parts are equal.

Common mistake

Do not base a proof on AAA or SSA; they do not work.

Shortcut

Order for proofs: 1) write the equal pairs. 2) name the test. 3) state the congruence. 4) conclude the equal parts. Memory clue: Never forget to end the proof with the line "corresponding parts are equal".

Spot the difference

PointAASAAA
What is equaltwo angles + one sidethree angles
Does it workyesno
Sizeexactly the samecan be bigger or smaller
Example60°, 70°, 8 cm60°, 60°, 60° (sides 3 and 6)

Practice questions

Easy
  1. 1Two triangles have sides 6, 8, 10 each. Which test?
  2. 2ΔABC ≅ ΔPQR, AC = 9 cm, ∠C = 35°. Find PR and ∠R.
  3. 3Which of SSS, SAS, AAA, ASA is not a congruence test?
Medium
  1. 4Two triangles have equal sides 5 cm and 7 cm with a 50° angle between them. Which test?
  2. 5ΔABC: ∠A = 40°, ∠B = 80°, AB = 6 ; ΔDEF: ∠D = 40°, ∠E = 80°, DE = 6. Which test?
  3. 6Two right triangles have hypotenuse 10 and one side 6. Find the other side and name the test.
Challenging
  1. 7Segments AB and CD bisect each other at O. Prove AC = BD (name the test).
  2. 8In ΔABC, AB = AC and AD ⟂ BC. Why is BD = DC?
Show answers (try them yourself first)
  1. 1SSS.
  2. 2PR = 9 cm, ∠R = 35°.
  3. 3AAA.
  4. 4SAS (the angle is between the sides).
  5. 5ASA (AB and DE lie between the two angles).
  6. 6The other side = √(100 − 36) = 8 ; test RHS.
  7. 7OA = OB, OC = OD, ∠AOC = ∠BOD (vertically opposite) → by SAS ΔAOC ≅ ΔBOD → AC = BD.
  8. 8In ΔABD and ΔACD: hypotenuse AB = AC, side AD common → congruent by RHS → BD = DC.