676re5weqwq

• Triangles are classified by their sides and angles.

Classification by Sides

• Scalene: All sides and angles have different measures.
• Isosceles: Two sides are equal and the angles opposite these sides are also equal.
• Equilateral: All three sides are equal, and each angle measures 60°.

Classification by Angles

• Acute: All three interior angles are less than 90°.
• Obtuse: One interior angle is greater than 90°.
• Right-angled: One interior angle is 90°.
• Triangles can have combined properties, like obtuse isosceles or right-angled isosceles.

Interior Angles of a Triangle

• The sum of the interior angles in any triangle is 180°.

Exterior Angles of a Triangle

• An exterior angle equals the sum of the two opposite interior angles.

Congruency

• Congruent triangles are exactly the same, with equal corresponding angles and sides.
• The symbol for congruency is ( \equiv ).

Congruency Rules

• RHS (Right-angle-Hypotenuse-Side) or 90°HS: The hypotenuse and one side of a right-angled triangle are equal to those of another right-angled triangle.
• SSS (Side-Side-Side): All three sides are equal to the corresponding sides of another triangle.
• SAS (Side-Angle-Side) or S∠S: Two sides and the included angle are equal to those of another triangle.
• AAS (Angle-Angle-Side) or ∠∠S: One side and two angles are equal to the corresponding side and angles of another triangle.
• The order of letters is important when labeling congruent triangles (e.g., ( \triangle ABC \equiv \triangle DEF )).

Similarity

• Similar triangles are scaled versions of each other with equal corresponding angles and proportional corresponding sides.
• The symbol for similarity is (\sim).

Similarity Rules

• AAA (Angle-Angle-Angle): All three pairs of corresponding angles are equal.
• SSS (Side-Side-Side): All three pairs of corresponding sides are in proportion.

Theorem of Pythagoras

• In a right-angled triangle, the square of the hypotenuse ((b)) is equal to the sum of the squares of the other two sides ((a) and (c)): [ b^2 = a^2 + c^2 ]
• Converse: If ( b^2 = a^2 + c^2 ), then the triangle is right-angled.

Quadrilaterals

Definition

• A quadrilateral is a closed shape with four straight line segments.
• The interior angles of any quadrilateral sum to 360°.

Parallelogram

• A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Properties of a Parallelogram

• Both pairs of opposite sides are parallel and equal in length.
• Both pairs of opposite angles are equal.
• Diagonals bisect each other.

Rectangle

• A rectangle is a parallelogram with all four angles equal to 90°.

Properties of a Rectangle

• Both pairs of opposite sides are parallel and equal in length.
• Both pairs of opposite angles are equal.
• All interior angles are equal to 90°.
• Diagonals bisect each other and are equal in length.

Rhombus

• A rhombus is a parallelogram with all four sides of equal length.

Properties of a Rhombus

• Both pairs of opposite sides are parallel and equal in length.
• Both pairs of opposite angles are equal.
• All sides are equal in length.
• The diagonals bisect each other at 90° and bisect both pairs of opposite angles.

Square

• A square is a rhombus with all four interior angles equal to 90°, or a rectangle with all four sides equal in length.

Properties of a Square

• Both pairs of opposite sides are parallel and equal in length.
• Both pairs of opposite angles are equal.
• All sides are equal in length.
• The diagonals bisect each other at 90°, bisect both pairs of opposite angles, and are equal in length.
• All interior angles equal 90°.

Trapezium

• A trapezium (or trapezoid) is a quadrilateral with one pair of opposite sides parallel.

Kite

• A kite is a quadrilateral with two pairs of adjacent sides equal.

Properties of a Kite

• The diagonal between equal sides bisects the other diagonal.
• One pair of opposite angles are equal (angles between unequal sides).
• The diagonal between equal sides bisects the interior angles and is an axis of symmetry.
• Diagonals intersect at 90°.

Relationships Between Different Quadrilaterals

• A square is both a rhombus and a rectangle.
• Rhombuses and rectangles are types of parallelograms.
• A parallelogram is a type of quadrilateral.
• Trapeziums and kites are quadrilaterals, but not necessarily parallelograms.

Visual Representation of Relationships

• Parallelogram: Includes rhombus and rectangle.
• Rectangle: Special type of parallelogram with right angles.
• Square: Special type of rectangle with equal sides.
• Rhombus: Special type of parallelogram with equal sides.
• Trapezium: A quadrilateral with one pair of parallel sides.
• Kite: A quadrilateral with two pairs of adjacent sides equal.

The Mid-Point Theorem

Definition

• The Mid-Point Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is equal to half the length of the third side.

Converse

• If a line is drawn through the mid-point of a side of a triangle parallel to the second side, it bisects the third side.

Properties

• Parallel Lines: The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
• Half-Length: This line segment is equal to half the length of the third side.

Applications

• Geometric Proofs: Used to prove parallelism and segment proportions in geometric figures.
• Coordinate Geometry: Helpful in calculating mid-points and distances in coordinate plane problems.
• Similarity: Often used in proving similarity in triangles and other geometric shapes.

Proofs and Conjectures

Parallelogram ABCD

• In parallelogram ABCD, the bisectors of the angles (AW, BX, CY, and DZ) have been constructed, you are also given (AB = CD), (AD = BC), (AB \parallel CD), (AD \parallel BC), (A\hat{} = C\hat{}), (B\hat{} = D\hat{}).
• Prove that MNOP is a parallelogram.

Proof that (M\hat{}2 = O\hat{}2)

In (\triangle CDZ) and (\triangle ABX):

[ DCZ\hat{} = BAX\hat{} \quad (\text{given}) ] [ D\hat{}1 = B\hat{}1 \quad (\text{given}) ] [ DC = AB \quad (\text{given}) ] [ \therefore \triangle CDZ \equiv \triangle ABX \quad (\text{AAS}) ] [ \therefore CZ = AX \quad \text{and} \quad CZD\hat{} = AXB\hat{} ]

In (\triangle XAM) and (\triangle ZCO):

[ XAM\hat{} = ZCO\hat{} \quad (\text{given: } \triangle CDZ \equiv \triangle ABX) ] [ AXM\hat{} = CZO\hat{} \quad (\text{proved above}) ] [ AX = CZ \quad (\text{proved above}) ] [ \therefore \triangle XAM \equiv \triangle ZCO \quad (\text{AAS}) ] [ \therefore M\hat{}1 = O\hat{}1 ] but (M\hat{}1 = M\hat{}2) (vert opp (\angle)s (=)) and (O\hat{}1 = O\hat{}2) (vert opp (\angle)s (=)) [ \therefore M\hat{}2 = O\hat{}2 ]

Conclusion

• Both pairs of opposite angles of MNOP are equal, therefore MNOP is a parallelogram.