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Lesson 3, Topic 2
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# Parallelograms & Triangles between Parallels

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In the diagram above, the four shapes lie between the same parallels $$\scriptsize \bar{UV}$$ and $$\scriptsize \bar{XY}$$ i.e. $$\scriptsize \bar{UV} || \bar{XY}$$, the parallelograms ABCD and EFGH have their bases on $$\scriptsize \bar{XY}$$ and their opposite sides on $$\scriptsize \bar{UV}$$, also the triangles JKL and MNO are drawn with their bases on $$\scriptsize\bar{XY}$$and their opposite vertices on $$\: \scriptsize \bar{UV}$$.

Note: that the altitudes of the four shapes are all equal (h). The altitude is the distance between the two parallel lines.

The following theorems have been proven to be true for Parallelograms and Triangles.

i. Parallelograms on the same base and between the same parallels are equal in area.

i.e. Area of ABCD = Area of PQCD.

ii. Triangles on the same base and between the same parallels are equal in area.

i.e. Area of ADP = Area of BCQ

### Example 1:

Name a triangle equal in area to the shaded triangle in each of the diagrams below:

(i)

Answer: $$\scriptsize \Delta PQR \: (Same\: base)$$

(ii)

Answer: $$\scriptsize \Delta ABD \\ \scriptsize (a)\: \Delta BCE = \Delta BDE, \: Same \:base \: i.e \: BE \\ \scriptsize (b)\: \Delta ABE \: common \: to \: both \: \Delta s \: BCE \: and \: BDE$$

(iii)

Answer: $$\scriptsize \Delta KLO \\ \scriptsize (a)\: \Delta KLN = \Delta KMN, \: Same \:base \: I.e \: KN \\ \scriptsize (b)\: \Delta KNO\: is \: common \: to \: both \: \Delta s \: KLN \: and \: KMN$$

(iv)

Answer: $$\scriptsize \Delta PQT \: (Same \: base \: i.e \: PQ)$$

### Example 2:

In the diagram below, name a quadrilateral equal in area to  $$\scriptsize \Delta PTR$$

Solution:

i. $$\scriptsize \Delta RTQ = QST \: (Same\: base \: i.e \: \bar{TQ})$$

ii. $$\scriptsize \Delta RTQ \: + \: \Delta PQT = \Delta QST \: + \: \Delta PQT \\ \scriptsize \Delta PTR = quadrilateral \: PQST$$

### Example 3:

In the rhombus KTMP, |XT| = |TS| . Name the parallelogram equal in area to RSMP. (WAEC)

Solution:

Parallelogram KPXY

Evaluation:

1. The angles of a quadrilateral, taken in order are x, 5x, 4x and 2x.
(a) Find these angles.
(b) Draw a rough sketch of the quadrilateral.
(c) What kind of quadrilateral is it?

2. Find the interior angles of a regular polygon which has (i) 6  (ii) 10 (iii) 20 sides.

3. Find the nearest degrees, the size of each angle of a regular heptagon (seven sides).

4. A regular polygon has angles of size 150o each. How many sides have the polygon?

5. Four angles of a pentagon are equal and the fifth is 60o. Find the equal angles and show that two sides of the pentagon are parallel.

6. In the diagram below ABCD and ABEC are parallelograms. EBF, DAF are straight lines. Prove that: