Explain why constructing a triangle with angles 60°, 70°, and 60° results in an isosceles triangle.

Because the sum of angles is 190°

Because it's an equilateral triangle

Quizzes / CBSE / Class 8 / Mathematics / Practical Geometry

Special Cases of Construction Quiz - CBSE Class 8 Mathematics

Free practice quiz: Special Cases of Construction (CBSE Class 8 Mathematics). 10 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 10 multiple-choice questions on Special Cases of Construction, and 10 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Start quiz: 10 questions, free

Learn the key ideas first ↓

Learn the key ideas Beta

Triangle Inequality Property

A triangle can be drawn only if the sum of any two sides is greater than the third side.
Quick check: add the two shorter sides and see if the total is more than the longest side.

Condition for sides a, b, c

\[a + b > c,\quad b + c > a,\quad c + a > b\]

Shortcut when c is the longest side

\[a + b > c\]

Common mistake: Accepting sides when the two shorter sides add up exactly to the longest. Equal is not enough; the sum must be strictly greater.

Criteria for a Unique Triangle

A triangle is fixed uniquely by SSS, SAS, ASA or RHS data.
For SAS, the given angle must lie between the two given sides (the included angle).
Base and altitude alone do not fix a triangle. The third vertex can be anywhere on a parallel line, giving infinitely many triangles.

SSS

\[\text{three sides}\]

SAS

\[\text{two sides} + \text{included angle}\]

ASA

\[\text{two angles} + \text{included side}\]

RHS

\[\text{right angle} + \text{hypotenuse} + \text{one side}\]

Common mistake: Using SAS when the given angle is not between the two given sides. That data may not give a unique triangle.

Angle Sum Property

The three angles of any triangle add up to 180°, and each angle must be greater than 0°.
Two angles fix the third angle but not the size. Many similar triangles fit, so one side length is also needed.

Angle sum

\[\angle A + \angle B + \angle C = 180^\circ\]

Third angle

\[\angle C = 180^\circ - \angle A - \angle B\]

Common mistake: Thinking three angles alone (AAA) fix a unique triangle. They fix only the shape; the size can vary.

Isosceles and Equilateral Triangles

Equal sides are opposite equal angles, and equal angles are opposite equal sides.
If all three angles are equal, all sides are equal, so the triangle is equilateral.

Isosceles property

\[AB = AC \iff \angle C = \angle B\]

Apex angle if each base angle is x

\[180^\circ - x - x\]

Equilateral

\[AB = BC = CA \iff \angle A = \angle B = \angle C\]

Common mistake: Matching an equal angle to the wrong side. An equal angle is always opposite its equal side, not next to it.

Right Triangles and Pythagoras

The hypotenuse is the side opposite the right angle and is always the longest side.
Use Pythagoras to find a missing side when two sides of a right triangle are known.

Pythagoras theorem (c is the hypotenuse)

\[a^2 + b^2 = c^2\]

Missing leg

\[b = \sqrt{c^2 - a^2}\]

Common mistake: Adding the squares when finding a leg. For a leg, subtract: hypotenuse squared minus the known leg squared.

Exam tips

Always draw a rough sketch first and mark the given sides and angles before constructing.
Check the triangle inequality and the angle sum before you start drawing.
Name the criterion used (SSS, SAS, ASA, RHS) when asked about the construction method.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

Construct a triangle with sides 4cm, 6cm, and 8cm. Which special case of triangle construction does this represent?
Show answer
Answer: C. Scalene triangle
A scalene triangle has all three sides of different lengths. Since the given sides (4cm, 6cm, 8cm) are all different, it's a scalene triangle.
Construct an isosceles triangle with sides 5cm and 5cm and a base angle of 60°. What type of special triangle is also created?
Show answer
Answer: B. Equilateral triangle
Since the two sides are equal and the base angle is 60°, the other base angle will also be 60° (due to isosceles triangle properties). The sum of angles in a triangle is 180°, so the third angle is also 60°. Thus, all angles are 60° forming an equilateral triangle.
You are given only the base and the corresponding altitude of a triangle. Is it possible to construct the triangle uniquely? If not, how many such triangles can be constructed?
Show answer
Answer: B. No, infinitely many triangles
With only the base and altitude, the third vertex can lie anywhere on a line parallel to the base at a distance equal to the altitude. This allows for infinite possibilities for the triangle's shape.
Explain why constructing a triangle with angles 60°, 70°, and 60° results in an isosceles triangle.
Show answer
Answer: B. Because two angles are equal
An isosceles triangle is defined as having at least two equal angles. The given angles have two 60° angles, making it an isosceles triangle. Note that it is also an equilateral triangle, but the question asks specifically for the property making it isosceles.
Construct a triangle with sides 5cm, 5cm and 6cm. What type of triangle is formed?
Show answer
Answer: B. Isosceles triangle
An isosceles triangle has at least two sides equal. Since two sides are 5cm each, it's an isosceles triangle.