Triangles, Quadrilateral & Polygons:
Chapter-11
Triangles:
Classification of Triangles:
1. Equilateral Triangle: A triangle with 3 equal sides
2. Isosceles triangle: A triangle with at least 2 equal sides
3. Scalene triangle: A triangle with no equal sides
* The sum of interior angles of a triangle is 180⁰
* An exterior angle of a triangle is equal to the sum of its interior opposite angles.
Q. 1. a) In the figure, ABD is a straight line.
Calculate the value of a.
Solution:
a) a⁰ = 56⁰ + 50⁰ (ext. ∠ of △)
= 106⁰
a = 106
b) In the figure, ACE, BCD and DEF are straight lines. Calculate the value of b and of c.
b) ACB = 94⁰ (vert. opp. ∠s)
b⁰ + 38⁰ + 94⁰ = 180⁰ (∠sum of △ ABC)
b⁰ = 180⁰ - 38⁰ - 94⁰
= 48⁰
c⁰ = 25⁰ + 94⁰
= 119 ( ext. ∠ of △CDE)
Practice:
a) In the figure, ABD is a straight line. Find the value of a.
Solution:
a) a⁰ = 53⁰ + 48⁰ (ext. ∠ of △)
= 101⁰
a = 101
b) In the figure, ABC, ADF and BDE are straight lines. Find the value of b and of c.
b) EDF = 93⁰ (vert. opp. ∠s)
b⁰ + 33⁰ + 93⁰ = 180⁰ (∠sum of △ ABC)
b⁰ = 180⁰ - 33⁰ - 93⁰
= 54⁰
c⁰ = 41⁰ + 93⁰
= 134 ( ext. ∠ of △CDE)
4. Angles in a Parallelogram:
Q. The figure shows a parallelogram ABCD where BAD = 64⁰. E lies on AB such that ADE = 49⁰.
Calculate
i) ABC
ii) CDE.
Solution:
i) ABC + 64⁰ = 180⁰ (int. ∠s, AD ∥ BC)
ABC = 180⁰ - 64⁰
= 116⁰
ii) ADC = 116⁰ (opp. ∠s, of ∥ gram)
CDE + 49⁰ = 116⁰
CDE = 116⁰ - 49⁰
= 67⁰
1. The figure shows a parallelogram ABCD where ADC = 108⁰. E lies on AB such that BCE = 38⁰.
i) Given that ABC = 9x⁰, find the value of x.
ii) Find DCE.
2. The figure shows a parallelogram ABCD. Find the value of x and of y.
Angles in a Rhombus:
The figure shows a rhombus ABCD. The diagonal BD is produced to E such that AD = DE.
If ABE = 68⁰, calculate
i) BCD
ii) DAE.
i) CBD = 68⁰ (diagonals bisect interior angles of a rhombus)
BCD + 68⁰ + 68⁰ = 180⁰ (int. ∠s, AB ∥ DC)
BCD = 180⁰ - 68⁰ - 68⁰
= 44⁰
ii) ADB = 68⁰ (base ∠Ñ• of isos. ΔABD)
DAE + AED = 68⁰ ext. ∠ of Δ)
DAE = 68/2
=34⁰
* Base angles of an isosceles triangle are equal.
Practice:
1. The figure shows a rhombus ABCD where ACD = 32. AB is produced to E such that AC = CE.
Find
i) ABC
ii) BCE
2. The figure shows a rhombus ABCD where the diagonals AC and BD intersect at E. Find the value of x.
Construction of Quadrilateral:
Quadrilateral is a closed plane figure that has 4 sides, 4 verticales, 4 interior angles.
Any point on the perpendicular bisector of a line segment is equidistant from the two end points of the line segment.
Q.1. Construct a quadrilateral PQRS such that PQ = 6 cm, QR = 7.5 cm, RS = 8.2 cm and the diagonal PR = 9.2 cm. Measure and write down the size of QRS.
Solution:
Construction Steps:
a) Using a ruler, draw PR = 9.2 cm.
b) Since S is 5.8 cm away from P, with P as centre and 5.8 cm as radius, draw arc 1.
c) Since S is 8.2 cm away from R, with R as centre and 8.2 cm as radius, draw arc 2 to cut arc 1 at S.
d) Join PS and RS.
e) Since Q is 6 cm away from P, with P as centre and 6 cm as radius, draw arc 3.
f) Since Q is 7.5 cm away from R, with R as centre and 7.5 cm as radius, draw arc 4 to cut arc 3 at Q.
g) Join PQ and QR.
QRS = 79
