# Trapezoid and its Theorems

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Trapezoid is a quadrilateral with at least one pair of parallel sides. AB || CD. (if there are two pairs of parallel lines then it is a parallelogram)

When non-parallel sides in trapezoid are equal then it is know ans isosceles trapezoid.

**Theorem 1: A trapezoid is isosceles if and only if the base angles are congruent.**

**Given :**ABCD is an isosceles trapezoid. AD = BC and AB || CD.

**Prove that :**∠C = ∠D

Statements |
Reasons |

1) ABCD is a trapezoid. | 1) Given |

2) AB || CD | 2) Given |

3) AD = BC | 3) Given |

4) DA || CE | 4) By construction |

5) ADCE is a parallelogram. | 5) By Properties of parallelogram. |

6) DA = CE and DC = AE | 6) By properties of parallelogram. |

7) BC = CE | 7) BC = AD and AD = CE (Transitive property) |

8) ∠CEB ≅ &CBE | 8) If BC ≅ CE then angle opposite to them are congruent. |

9) ∠DAB ≅ ∠ABC | 9) property of parallelogram and linear pair angles |

10) ∠A + ∠D = 180 and ∠B + ∠C = 180 | 10) Interior angles on the same side of the transversal are supplementary. |

11) ∠A + ∠D = ∠C + ∠B | 11) Transitivity ( Right sides are same so left sides are equal) |

12) ∠D = ∠C | 12) From above (∠A = ∠B) |

**Example :**In a trapezoid PQRS, PQ||RS and PS = QR. If ∠S = 60

^{0}then find the remaining angles.

**Solution :**

PQ||RS and PS = QR, so trapezoid PQRS is an isosceles trapezoid.

In isosceles trapezoid, base angles are equal.(trapezoid and its theorems)

∠S = ∠R and ∠P = ∠Q

But ∠S = 60

^{0}

∴ ∠R = 60

^{0}

Let ∠P = ∠Q = x

Sum of all the angles in a quadrilateral is 360.

∴ ∠P + ∠Q + ∠S + ∠R = 360

x + x + 60 + 60 = 360

2x +120 = 360

2x = 360 -120

2x = 240

∴ x = 240/2

x = 120

∠P = ∠Q = 120

^{0}

__________________________________________________________________

Some important theorems of trapezoids are given below :

Theorems |

1. A trapezoid is isosceles if and only if the base angles are congruent. |

2. A trapezoid is isosceles if and only if the diagonals are congruent. |

3. If a trapezoid is isosceles, the opposite angles are supplementary. |

The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. |

Never assume that a trapezoid is isosceles unless you are given (or can prove) that information. |

**Practice**

1) In a trapezoid ABCD,AB|| CD and BC = AD. If m∠C=65

^{0}then find m∠D.

2) PQRS is a trapezium in which PQ || RS. If ∠P = ∠Q = 40, find the measures of other two angles.

3) In trapezoid ABCD, ∠B= 120

^{0}Find m∠C.

4) In a quadrilateral HELP, if EP = LH then what type of quadrilateral it is?

5) In a quadrilateral, the angles are in the ratio of 4:5:3:6.Find the measures of each angles.

6) If three angles in the trapezoid are 130

^{0},120

^{0},50

^{0}and 2x

^{0}. Find x and the 4th angle.

7) Draw a isosceles trapezoid named PQRS, PS||QR and PQ = SR.

**Quadrilateral**

• Introduction to Quadrilateral

• Types of Quadrilateral

• Properties of Quadrilateral

• Parallelogram and its Theorems

• Rectangle and its Theorems

• Square and its Theorems

• Rhombus and its Theorems

• Trapezoid and its Theorems

• Kite and its Theorems

• Mid Point Theorem

• Introduction to Quadrilateral

• Types of Quadrilateral

• Properties of Quadrilateral

• Parallelogram and its Theorems

• Rectangle and its Theorems

• Square and its Theorems

• Rhombus and its Theorems

• Trapezoid and its Theorems

• Kite and its Theorems

• Mid Point Theorem

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