Std 6 Mathematics · Chapter 8

8. Triangles and Squares

Solutions to Practice Set 8.1, 8.2, 8.3 and Exercise 8. Maharashtra State Board (Balbharati), Grade Six Mathematics.

About the chapter

A triangle is a closed figure made by three line segments. A quadrilateral is a closed figure made by four line segments. In this chapter, we learn the vertices, sides and angles of these figures. We also learn the types of triangles and the parts of a quadrilateral.

Key points

A triangle has 3 vertices, 3 sides and 3 angles.
By sides, a triangle can be equilateral, isosceles or scalene.
By angles, a triangle can be acute angled, right angled or obtuse angled.
A quadrilateral has 4 vertices, 4 sides, 4 angles and 2 diagonals.
A quadrilateral can be named in 8 ways. We go around its vertices in order.

Formulas and terms

Equilateral triangle – all three sides are equal.
Isosceles triangle – two sides are equal. Scalene triangle – all sides are of different lengths.
Adjacent sides – sides of a quadrilateral that have a common vertex.
Opposite sides – sides of a quadrilateral that have no common vertex.
Diagonal – the segment joining the vertices of opposite angles.

Practice Set 8.1

1Complete the table by observing the diagrams.

△ABC

△LMN

△XYZ

Sr. No.	Figure	Vertices	Sides	Angles
1	△ABC	A, B, C	AB, BC, AC	∠A, ∠B, ∠C
2	△LMN	L, M, N	LM, MN, LN	∠L, ∠M, ∠N
3	△XYZ	X, Y, Z	XY, YZ, XZ	∠X, ∠Y, ∠Z

2Observe the figure and write the names of all the triangles.

D lies on AB, E lies on AC, and the segments BE and CD cross at F.

There are 12 triangles in the figure:

△ABC, △ADE, △ABE, △ACD, △BCD, △BCE, △BCF, △BDE, △CDE, △DEF, △BDF, △CEF

3Observe the figure and answer the questions.

F and E lie on the triangle; A, B, C are inside it; G and D are outside it.

(i) Name of the triangle – △PQR
(ii) Points on the triangle – Point F, Point E (and the vertices P, Q, R)
(iii) Points in the interior of the triangle – Point A, Point B, Point C
(iv) Points in the exterior of the triangle – Point G, Point D
(v) Points in the triangular region – P, Q, R, F, E, A, B, C

The triangular region means the triangle and its interior together.

Practice Set 8.2

1Equal sides are shown by the same marking. Write the type of triangle.

(i) △DEF – All three sides are equal. So, it is an equilateral triangle.
(ii) △MAN – Two sides are equal. So, it is an isosceles triangle.
(iii) △EFG – All sides are of different lengths. So, it is a scalene triangle.

2Write the type of triangle from the information given.

(i) In △ABC, m∠A = 63°, m∠B = 52°, m∠C = 65°. – Acute angled triangle
All three angles are less than 90°.
(ii) In △RST, seg RS and seg ST are equal. – Isosceles triangle
(iii) In △LMN, ∠LMN is a right angle. – Right angled triangle
(iv) In △SDO, m∠OSD = 135°. – Obtuse angled triangle
One angle is more than 90°.
(v) All the sides of △RTM are equal. – Equilateral triangle
(vi) The sides of △DEF are 5.3 cm, 4.2 cm and 6.6 cm. – Scalene triangle

Practice Set 8.3

1Observe □XYZW and answer the following questions.

□XYZW — the dashed lines are the two diagonals.

(i) Which side is opposite to side XY? – Side WZ
(ii) Which are the adjacent sides of side XW? – Side XY and side WZ
(iii) Which is the opposite angle of ∠W? – ∠Y
(iv) Which are the adjacent angles of ∠Z? – ∠Y and ∠W
(v) Which are the diagonals of □XYZW? – Diagonal XZ and diagonal YW

2Write the opposite angles in □PQRS.

The pairs of opposite angles in □PQRS are:

∠P and ∠R
∠Q and ∠S

Exercise 8

1Fill in the blanks.

(i) A triangle in which all three sides have the same length is called an equilateral triangle.
(ii) A triangle in which only two sides have equal lengths is called an isosceles triangle.
(iii) A triangle whose one angle is 90 degrees is called a right angled triangle.
(iv) The line segment joining the vertices of the opposite angles of a quadrilateral is called a diagonal.
(v) In □LMNO, side LM and side MN are adjacent sides. (Side OL is also correct.)

2State whether true or false.

(i) A triangle can have two right angles. – False
Two right angles make 180°. Nothing is left for the third angle.
(ii) A right angled triangle can be an isosceles triangle. – True
(iii) An acute-angled triangle cannot be an equilateral triangle. – False
Each angle of an equilateral triangle is 60°. So it is always acute angled.

3Draw □UVWX and answer the questions from it.

(i) Names of the quadrilateral: □UVWX, □VWXU, □WXUV, □XUVW, □UXWV, □XWVU, □WVUX, □VUXW
(ii) Pairs of opposite sides: side UV and side XW; side VW and side UX
(iii) Pairs of adjacent sides: UV and VW; VW and WX; WX and XU; XU and UV
(iv) Diagonals: diagonal UW and diagonal VX
(v) Pairs of opposite angles: ∠U and ∠W; ∠V and ∠X
(vi) Pairs of adjacent angles: ∠U and ∠V; ∠V and ∠W; ∠W and ∠X; ∠X and ∠U

4How many triangles are there in each figure?

(i) — 3 triangles

(ii) — 6 triangles

(iii) — 7 triangles

(iv) — 10 triangles

Figure (i) – 3 triangles
Figure (ii) – 6 triangles
Figure (iii) – 7 triangles
Figure (iv) – 10 triangles

Note on (iii): If the extra line in your book goes fully across the big triangle, count again. Then the answer is 12.

5Count and write the total number of quadrilaterals in the figure.

A grid of 4 rows and 5 columns of squares.

A quadrilateral is formed by any 2 horizontal lines and any 2 vertical lines.

Horizontal lines = 5. Pairs of them = (5 × 4) ÷ 2 = 10
Vertical lines = 6. Pairs of them = (6 × 5) ÷ 2 = 15
Total quadrilaterals = 10 × 15 = 150

There are 150 quadrilaterals in the figure.

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