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Step-by-step explanation:
1.
Angle Sum Property of a Quadrilateral
According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees.
Quadrilateral: Angle Sum Property
Proof: In the quadrilateral ABCD,
∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.
AC is a diagonal
AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC
We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.
We know that the sum of angles in a triangle is 180°.
Now consider triangle ADC,
∠D + ∠DAC + ∠DCA = 180° (Sum of angles in a triangle)
Now consider triangle ABC,
∠B + ∠BAC + ∠BCA = 180° (Sum of angles in a triangle)
On adding both the equations obtained above we have,
(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°
∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°
We see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD.
Replacing them we have,
∠D + ∠DAB + ∠BCD + ∠B = 360°
That is,
∠D + ∠A + ∠C + ∠B = 360°.
Or, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.
Quadrilateral Angles
A quadrilateral has 4 angles. The sum of its interior angles is 360 degrees. We can find the angles of a quadrilateral if we know 3 angles or 2 angles or 1 angle and 4 lengths of the quadrilateral. In the image given below, a Trapezoid (also a type of Quadrilateral) is shown.
The sum of all the angles ∠A +∠B + ∠C + ∠D = 360°
Quadrilateral Angles
In the case of square and rectangle, the value of all the angles is 90 degrees. Hence,
∠A = ∠B = ∠C = ∠D = 90°
A quadrilateral, in general, has sides of different lengths and angles of different measures. However, squares, rectangles, etc. are special types of quadrilaterals with some of their sides and angles being equal.
2.
There are six basic types of quadrilaterals. They are:
Trapezium
Parallelogram
Rectangle
Rhombus
Square
Kite
3.
There are six important properties of parallelograms to know:
Opposite sides are congruent (AB = DC).
Opposite angels are congruent (D = B).
Consecutive angles are supplementary (A + D = 180°).
If one angle is right, then all angles are right.
The diagonals of a parallelogram bisect each other.
Each diagonal of a parallelogram separates it into two congruent triangles.
3.
MidPoint Theorem Statement
The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”
Mid- Point Theorem
MidPoint Theorem Proof
If midpoints of any of the sides of a triangle are adjoined by the line segment, then the line segment is said to be in parallel to all the remaining sides and also will measure about half of the remaining sides.
Consider the triangle ABC, as shown in the above figure,
Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the side BC, whereas the side DE is half of the side BC; i.e.
DE∥BC
DE = (1/2 * BC).
Now consider the below figure,
Mid- Point Theorem
Construction- Extend the line segment DE and produce it to F such that, EF=DE.
In the triangle, ADE, and also the triangle CFE
EC= AE —– (given)
∠CEF = ∠AED {vertically opposite angles}
EF = DE { by construction}
hence,
△ CFE ≅ △ ADE {by SAS}
Therefore,
∠CFE = ∠ADE {by c.p.c.t.}
∠FCE= ∠DAE {by c.p.c.t.}
and CF = AD {by c.p.c.t.}
The angles, ∠CFE and ∠ADE are the alternate interior angles. Assume CF and AB as two lines which are intersected by the transversal DF.
In a similar way, ∠FCE and ∠DAE are the alternate interior angles. Assume CF and AB are the two lines which are intersected by the transversal AC.
Therefore, CF ∥ AB
So, CF ∥ BD
and CF = BD {since BD = AD, it is proved that CF = AD}
Thus, BDFC forms a parallelogram.
By the use of properties of a parallelogram, we can write
BC ∥ DF
and BC = DF
BC ∥ DE
and DE = (1/2 * BC).
Hence, the midpoint theorem is Proved
Hope this helps!!!