# Congruent Triangles

See how to prove that the given triangles are congruent.

5 examples and their solutions.

## SSS Congruence

### Postulate

if 3 sides of each triangle are congruent

(Side-Side-Side),

then those two triangles are congruent.

### Example

Given: AB ≌ AD

C is the midpoint of BD.

Prove: △ABC ≌ △ADC

Solution C is the midpoint of BD.

Prove: △ABC ≌ △ADC

Statement | Reason |
---|---|

1. AB ≌ AD | Given |

2. C is the midpoint of BD. | Given |

3. BC ≌ CD | Definition of a midpoint |

4. AC ≌ AC | Reflexive property |

5. △ABC ≌ △ADC | SSS Congruence |

Close

## SAS Congruence

### Postulate

if 2 sides & 1 angle of each triangle are congruent

(Side-Angle-Side),

then those two triangles are congruent.

### Example

Given: P is the midpoint of AD and BC.

Prove: △ABP ≌ △DCP

Solution Prove: △ABP ≌ △DCP

Statement | Reason |
---|---|

1. P is the midpoint of AD and BC. | Given |

2. AP ≌ PD | Definition of a midpoint |

3. BP ≌ PC | Definition of a midpoint |

4. ∠APB ≌ ∠DCP | Vertical angles |

5. △ABP ≌ △DCP | SAS Congruence |

Close

## ASA Congruence

### Postulate

if 1 side & 2 angles of each triangle are congruent

(Angle-Side-Angle),

then those two triangles are congruent.

### Example

Given: AC bisects ∠BAD and ∠BCD.

Prove: △ABC ≌ △ADC

Solution Prove: △ABC ≌ △ADC

Statement | Reason |
---|---|

1. AC bisects ∠BAD and ∠BCD. | Given |

2. ∠BAC ≌ ∠DAC | Definition of an angle bisector |

3. ∠BCA ≌ ∠DCA | Definition of an angle bisector |

4. AC ≌ AC | Reflexive property |

5. △ABC ≌ △ADC | ASA Congruence |

Close

## AAS Congruence

### Theorem

if 2 angles & non-included side of each triangle are congruent

(Angle-Angle-Side),

then those two triangles are congruent.

### Example

Given: AB ≌ CD

∠PAB ≌ ∠PCD.

Prove: △PAB ≌ △PCD

Solution ∠PAB ≌ ∠PCD.

Prove: △PAB ≌ △PCD

Statement | Reason |
---|---|

1. AB ≌ CD | Given |

2. ∠PAB ≌ ∠PCD | Given |

3. ∠APB ≌ ∠CPD | Vertical Angles |

4. △ABC ≌ △ADC | AAS Congruence |

Close

## HL Congruence

### Postulate

if a hypotenuse and a leg of each triangle are congruent,

then those two right triangles are congruent.

### Example

Given: ∠A and ∠D are right angles.

AB ≌ CD

Prove: △ABC ≌ △DCB

Solution AB ≌ CD

Prove: △ABC ≌ △DCB

Statement | Reason |
---|---|

1. ∠A and ∠D are right angles. | Given |

2. △ABC and △DCB are right triangles. | Definition of a right triangle |

3. AB ≌ CD | Given |

4. BC ≌ BC | Reflexive property |

5. △ABC ≌ △DCB | HL Congruence |

2. To use the HL congruence,

first show that

the given triangles are right triangles.

first show that

the given triangles are right triangles.

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