Congruence
In geometry, two objects are congruent if one of them is placed on the other, and they exactly coincide. Congruent objects are equal in all respects. They have same shape, same size, and same area.
For example, two figures drawn on a piece of paper are congruent if they can be cut out and matched up perfectly (where flipping the paper over is allowed). The symbol used for congruence is \(\cong\).
For example, the sentence \(\triangle ABC \cong \triangle PQR\) is read as ‘triangle \(\triangle ABC\) is congruent to triangle \(\triangle PQR\).
The below shows the actual measurements of both triangles \(\triangle ABC\) and \(\triangle PQR\)
| Corresponding Sides | Corresponding Angles |
|---|---|
| AB = PQ | ∠A = ∠P |
| BC = QR | ∠B = ∠Q |
| AC = PR | ∠C = ∠R |
Congruence Axiom
Two traingles are congruent if following axioms are hold
- SSS
- ASA
- SAS
- AAS
- RHS
SSS
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
| Given | Results | ||
|---|---|---|---|
| Sides | Angles | Sides | Angles |
| AB=PQ | ∠A = ∠P | ||
| BC = QR | ∠B = ∠Q | ||
| AC = PR | ∠C = ∠R | ||
ASA
If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
| Given | Results | ||
|---|---|---|---|
| Sides | Angles | Sides | Angles |
| AB=PQ | ∠A = ∠P | ||
| ∠B = ∠Q | BC = QR | ||
| AC = PR | ∠C = ∠R | ||
SAS
If two sides and the included angle of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
| Given | Results | ||
|---|---|---|---|
| Sides | Angles | Sides | Angles |
| AB=PQ | ∠A = ∠P | ||
| AC=PR | BC = QR | ∠B = ∠Q | |
| ∠C = ∠R | |||
AAS
If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
| Given | Results | ||
|---|---|---|---|
| Sides | Angles | Sides | Angles |
| BC=QR | ∠A = ∠P | ||
| ∠B = ∠Q | AB=BC | ||
| AC=PR | ∠C = ∠R | ||
RHS
If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.
| Given | Results | ||
|---|---|---|---|
| Sides | Angles | Sides | Angles |
| BC=QR | ∠A = ∠P | ||
| AB=PQ | ∠B = ∠Q | ||
| AC=PR | ∠C = ∠R | ||
Test your Understanding: Quiz 1
Quiz Complete!
Your final score is:
Test your Understanding: Quiz 2
Drag and drop each term into the correct category.
Draggable Terms
Congruence
NOT Congruence
Test your Understanding: Quiz 3
Based on the information provided, what congruence theorem proves that △ABC ≅ △PQR?
No comments:
Post a Comment