Statements |
Reasons |

1) ΔABC ~ ΔPQR | 1) Given |

2) AB BC AC ------- = -------- = ------- PQ QR PR |
2) If two Δ's are similar then their corresponding sides are in the ratio |

3) ∠ABM = ∠PQN | 3) As ΔABC ~ ΔPQR |

4) ∠AMB = ∠PNQ | 4) Each of measure 90^{0} |

5) ΔAMB ~ ΔPNQ | 5) By AA similarity |

6) AM PN ------- = -------- AB PQ |
6) If two Δ's are similar then their corresponding sides are in the ratio |

7) AM PN ------- = -------- BC QR |
7) From (2) and (6) |

8) Area(ΔABC) (Base x height)/2 -------------- = ------------------- Area(ΔABC) (Base x height)/2 |
8) By definition of Area |

9) Area(ΔABC) (BC x AN) ----------- = --------------- Area(ΔABC) (QR x PN) |
9) By substitution |

10) Area(ΔABC) (BC x BC) ------------- = -------------- Area(ΔABC) (QR x QR) |
10) By substitution |

11) Area(ΔABC) BC^{2}---------- = ---------- Area(ΔABC) QR ^{2} |
11) By property of multiplication |

Statements |
Reasons |

1) Area(ΔABC) AB^{2}--------------- = ---------- Area(ΔDEF) DE ^{2} |
1) The ratio of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides. |

2) ∠A = ∠D | 2) As the two triangles are similar |

3) 1/2∠A = 1/2∠D | 3) Multiply both sides by 1/2 |

4) ∠BAX = ∠EDY | 4) By definition of angle bisector |

5) ∠B = ∠E | 5) ΔABC ~ ΔDEF |

6) ΔABX ~ ΔDEY | 6) By AA criteria or rule |

AB AX 7) ------- = -------- DE DY |
7)If two triangles are similar then their corresponding sides are in ratio. |

AB^{2} AX^{2}8) ------- = -------- DE ^{2} DY^{2} |
8) Squaring both sides |

Area(ΔABC) AX^{2}9) --------------- = ---------- Area(ΔDEF) DY ^{2} |
9) From (1) and (8) |

•

• GeometryProof-1

• GeometryProof 2

• area-similartriangles

• Pythagorean theorem

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