Download GEOMETRY II 20 MAY 2013 Key Concepts

GEOMETRY II
Lesson Description
In this lesson, we:
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
Look at the properties of the different quadrilaterals
Revisit similarity and congruency
Key Concepts
Properties of Quadrilaterals
Parallelogram
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are equal in length.
Both pairs of opposite angles are equal.
Both diagonals bisect each other.
Rectangle
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are equal in length.
Both pairs of opposite angles are equal.
Both diagonals bisect each other.
Diagonals are equal in length.
All angles are right angles.
Rhombus
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are equal in length.
Both pairs of opposite angles are equal.
Both diagonals bisect each other.
All sides are equal in length.
Die diagonals bisect each other at 90°.
Die diagonals bisect both pairs of opposite
angles.
20 MAY 2013
Square
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are equal in length.
Both pairs of opposite angles are equal.
Both diagonals bisect each other.
All sides are equal in length.
Die diagonals bisect each other at 90°.
Die diagonals bisect both pairs of opposite
angles.
All interior angles equal 90°.
Diagonals are equal in length.
Diagonals bisect both pairs of interior opposite
angles (i.e. all are 45°).
Trapezium
A trapezium is a quadrilateral with one pair of opposite sides parallel.
Kite
Diagonal between equal sides bisects the other
diagonal.
One pair of opposite angles are equal (the angles
between unequal sides).
Diagonal between equal sides bisects die interior
angles.
Diagonals intersect at 90°.
Questions
Question 1
ABCD is a parallelogram with diagonal AC.
Given that AF=HC, show that:
a.) △AFD≡△CHB
b.) DF∥HB
c.) DFBH is a parallelogram
Question 2
△PQR and △PSR are equilateral triangles. Prove that PQRS is a rhombus.
Question 3
Given parallelogram ABCD with AE and FC, AE bisecting
a.) Write all interior angles in terms of y.
b.) Prove that AFCE is a parallelogram.
and FC bisecting :
Question 4
Given that WZ=ZY=YX,
= and WX∥ZY, prove that:
a.) XZ bisects
b.) WY=XZ
Question 5
LMNO is a quadrilateral with LM=LO and diagonals that intersect at S such that MS=SO. Prove that:
a.) M S=S O
b.) △LON≡△LMN
c.) MO⊥LN
Question 6
Using the figure below, show that the sum of the three angles in a triangle is 180°. Line DE is parallel
to BC.
Question 7
D is a point on BC, in △ABC. N is the mid-point of AD. O is the mid-point of AB and M is the mid-point
of BD. NR∥AC.
a.) Prove that OBMN is a parallelogram.
b.) Prove that BC=2MR.
Question 8
PQR is an isosceles with PR=QR. S is the mid-point of PQ, T is the mid-point of PR and U is the midpoint of RQ.
a.) Prove △STU is also isosceles.
b.) What type of quadrilateral is STRU? Motivate your answer.
c.) If R U=68° calculate, with reasons, the size of T U.
Question 9
In △MNP, M=90°, S is the mid-point of MN and T is the mid-point of NR.
a.) Prove U is the mid-point of NP.
b.) If ST=4 cm and the area of △SNT is 6 cm 2, calculate the area of △MNR.
c.) Prove that the area of △MNR will always be four times the area of △SNT, let ST=x units and
SN=y units.