Geometry Theorems — Triangle Congruence Theorems
Geometry Theorems — Triangle Congruence Theorems. Practice questions to deepen understanding of triangle congruence theorems. Online math practice with full solutions and step-by-step explanations.
Triangle congruence practice — SAS, ASA, SSS, SSA in a right triangle. Identification and application in proofs.
Definition of congruence, SAS theorem (side-angle-side), ASA theorem (angle-side-angle).
△ Definition:
Two triangles are congruent if:
Two triangles are congruent if:
• All corresponding sides are equal
• All corresponding angles are equal
That is: one can be placed exactly on top of the other ✓
Notation: △ABC ≅ △DEF
If △ABC ≅ △DEF, then:
• AB = DE
• BC = EF
• AC = DF
• ∠A = ∠D
• ∠B = ∠E
• ∠C = ∠F
⭐ SAS:
Two triangles are congruent if:
If in two triangles:
• Two sides are equal
• The angle between them (included) is equal
then the triangles are congruent! ✓
SAS = Side-Angle-Side
The angle must be included (between the two sides)!
If the angle is not included — congruence is not guaranteed ✗
⭐ ASA:
Two triangles are congruent if:
If in two triangles:
• Two angles are equal
• The side between them (included) is equal
then the triangles are congruent! ✓
ASA = Angle-Side-Angle
The side must be included (between the two angles)!
If the side is not included — congruence is not guaranteed ✗
⭐ SSS:
Two triangles are congruent if:
If in two triangles:
• All three sides are equal
then the triangles are congruent! ✓
SSS = Side-Side-Side
Only three sides need to be checked ✓
No angles needed!
⭐ SSA special:
In a right triangle, two triangles are congruent if:
If in right triangles:
• Two sides are equal (hypotenuse+leg or two legs)
• The angle opposite the longer side is equal
then the triangles are congruent! ✓
This applies only to right triangles!
In right triangles:
• hypotenuse = hypotenuse
• leg = leg
• angle opposite the hypotenuse (90°) is equal
⇒ congruent! ✓
In a general triangle, SSA does not always imply congruence!
It works only in right triangles ✓
⚠️ Caution:
What does not necessarily imply congruence?
Three equal angles ⇏ congruence ✗
This implies similarity, not congruence!
The triangles may have the same shape but different sizes
✓ SSS
✓ SAS
✓ ASA
✓ SSA (right triangle)
✗ AAA (this is similarity!)
🔍 Identification:
In triangles ABC and DEF: AB=DE, AC=DF, ∠A=∠D. The triangles are:
• AB = DE (side)
• AC = DF (side)
• ∠A = ∠D (angle)
Check:
Angle A is between sides AB and AC ✓
so the angle is included!
Conclusion:
Congruence by SAS! ✓
🔍 Identification:
In triangles ABC and DEF: ∠A=∠D, AB=DE, ∠B=∠E. The triangles are:
• ∠A = ∠D (angle)
• AB = DE (side)
• ∠B = ∠E (angle)
Check:
Side AB is between angles A and B ✓
so the side is included!
Conclusion:
Congruence by ASA! ✓
🔍 Identification:
Two triangles have sides 5, 7, 9 cm each. The triangles are:
Triangle 1: sides 5, 7, 9 cm
Triangle 2: sides 5, 7, 9 cm
Check:
All three sides are equal! ✓
Conclusion:
Congruence by SSS! ✓
The simplest case!
△ Application:
To prove that the altitude to the base of an isosceles triangle divides it into two congruent triangles, we use:
Triangle ABC isosceles (AB=AC)
Altitude AD to base BC
Proof:
In triangles ABD and ACD:
1. AB = AC (legs) ✓
2. ∠ADB = ∠ADC = 90° (altitude) ✓
3. AD is common ✓
Conclusion:
Congruence by SAS!
(Side-Angle-Side)
❓ Condition:
To prove congruence by SAS, the angle must be:
In the SAS theorem:
The angle must be included!
That is: the angle is between the two sides ✓
If the angle is not included — congruence is not guaranteed! ✗
AB = DE
∠A = ∠D
AC = DF
Angle A is between AB and AC ✓
Congruence!
AB = DE
AC = DF
∠C = ∠F (not included!)
Angle C is not between AB and AC ✗
Congruence not guaranteed!
❓ Condition:
To prove congruence by ASA, the side must be:
In the ASA theorem:
The side must be included!
That is: the side is between the two angles ✓
If the side is not included — congruence is not guaranteed! ✗
∠A = ∠D
AB = DE
∠B = ∠E
Side AB is between ∠A and ∠B ✓
Congruence!
🔤 Order:
When writing congruence △ABC ≅ △DEF, the order:
When you write △ABC ≅ △DEF
the meaning is:
• A corresponds to D
• B corresponds to E
• C corresponds to F
The order must match! ✓
• AB = DE
• BC = EF
• AC = DF
• ∠A = ∠D
• ∠B = ∠E
• ∠C = ∠F
If △ABC ≅ △EDF
This means:
• A↔E, B↔D, C↔F
Not the same!
🔢 Computation:
If △ABC ≅ △DEF, AB=5 cm, BC=7 cm, AC=9 cm, then DE equals:
△ABC ≅ △DEF
AB = 5 cm
BC = 7 cm
AC = 9 cm
Solution:
From the congruence:
A corresponds to D
B corresponds to E
Therefore: AB = DE
DE = 5 cm ✓
🔢 Computation:
If △ABC ≅ △PQR, ∠A=40°, ∠B=60°, then ∠Q equals:
△ABC ≅ △PQR
∠A = 40°
∠B = 60°
Solution:
From the congruence:
A corresponds to P
B corresponds to Q
C corresponds to R
Therefore: ∠B = ∠Q
∠Q = 60° ✓
⚠️ Caution:
In two triangles: AB=DE, BC=EF, ∠A=∠D. Are they congruent?
AB = DE (side)
BC = EF (side)
∠A = ∠D (angle)
Check:
Is angle A included between AB and BC?
No! ✗
Angle A is between AB and AC, not BC!
Conclusion:
This is not SAS!
Congruence is not guaranteed! ✗
The angle is not between the two given sides
A common mistake! ⚠️
📐 Application:
To prove that two angles are equal, we can:
To prove that two angles are equal:
1. Find two triangles
2. Prove that they are congruent
3. Use congruence to conclude that the two corresponding angles are equal ✓
A very common method!
Want to prove: ∠ABC = ∠DEF
Strategy:
1. Prove △ABC ≅ △DEF
2. Therefore ∠ABC = ∠DEF (corresponding angles) ✓
📐 Application:
To prove that two sides are equal, we can:
To prove that two sides are equal:
1. Find two triangles
2. Prove that they are congruent
3. Use congruence to conclude that the two corresponding sides are equal ✓
Want to prove: AB = DE
Prove: △ABC ≅ △DEF
Therefore: AB = DE ✓
📏 Property:
If two triangles are congruent, then:
If △ABC ≅ △DEF, then:
• All sides are equal
AB=DE, BC=EF, AC=DF
• therefore the perimeters are equal! ✓
• the areas are also equal! ✓
• and all angles are equal ✓
If perimeter △ABC = 20 cm
and the triangles are congruent,
then perimeter △DEF = 20 cm ✓
📚 Summary:
Which is not a valid congruence theorem?
Three equal angles ⇏ congruence ✗
This implies similarity, not congruence!
Triangles may have the same shape but different sizes
✅ SSS — three equal sides
✅ SAS — two sides and the included angle
✅ ASA — two angles and the included side
✅ SSA — in right triangles: two sides and the angle opposite the longer
• In SAS: the angle must be included!
• In ASA: the side must be included!
• AAA = similarity, not congruence!