Context: In an equilateral triangle, each angle is . Q. Verify this by construction.

Understand the Question An equilateral triangle has all three sides of equal length (). To construct one, draw a base segment , then draw arcs of radius centered at and that intersect at point . Since all sides are equal, their opposite angles are also equal ($ A = B = C180^60^$. Step 1 · Construct the Equilateral Triangle 1. Draw a line segment of any length. 2. With center and radius equal to , draw an arc. 3. With center and the same radius , draw another arc intersecting the first arc at point . 4. Join and . Since all sides are equal to the radius : Thus, is an equilateral triangle. Step 2 · Verify the Angles In , all sides are equal: Angles opposite to equal sides are equal: Let . By the angle sum property of a triangle: Measuring with a protractor also confirms that each angle is . Answer > Each angle in an equilateral triangle is . Common Mistakes Changing Compass Radius: Changing the compass width while drawing the two arcs from and will make the sides unequal, resulting in a scalene or isosceles triangle instead of an equilateral triangle. Protractor Misalignment: Placing the center point of the protractor incorrectly on vertices , , or will give inaccurate angle readings.

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Question 32

Context: In an equilateral triangle, each angle is $60^\circ$ .

Q. Verify this by construction.

Solution

Understand the Question

An equilateral triangle has all three sides of equal length ( $AB = BC = AC$ ).
To construct one, draw a base segment $AB$ , then draw arcs of radius $AB$ centered at $A$ and $B$ that intersect at point $C$ .
Since all sides are equal, their opposite angles are also equal ( $\angle A = \angle B = \angle C$ ). By the angle sum property of a triangle ( $180^\circ$ ), each angle evaluates to $60^\circ$ .

Step 1 · Construct the Equilateral Triangle

Draw a line segment $AB$ of any length.
With center $A$ and radius equal to $AB$ , draw an arc.
With center $B$ and the same radius $AB$ , draw another arc intersecting the first arc at point $C$ .
Join $AC$ and $BC$ .

Since all sides are equal to the radius $AB$ : $AB = BC = AC$

Thus, $\Delta ABC$ is an equilateral triangle.

Step 2 · Verify the Angles

In $\Delta ABC$ , all sides are equal: $AB = BC = AC$

Angles opposite to equal sides are equal: $\angle A = \angle B = \angle C$

Let $\angle A = \angle B = \angle C = x$ .

By the angle sum property of a triangle:

\begin{aligned} \angle A + \angle B + \angle C &= 180^\circ \\ x + x + x &= 180^\circ \\ 3x &= 180^\circ \\ x &= \dfrac{180^\circ}{3} = 60^\circ \end{aligned}

Measuring with a protractor also confirms that each angle is $60^\circ$ .

Answer

Each angle in an equilateral triangle is $60^\circ$ .

Common Mistakes

Changing Compass Radius: Changing the compass width while drawing the two arcs from $A$ and $B$ will make the sides unequal, resulting in a scalene or isosceles triangle instead of an equilateral triangle.
Protractor Misalignment: Placing the center point of the protractor incorrectly on vertices $A$ , $B$ , or $C$ will give inaccurate angle readings.

More questions in IT

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Q. How do we do it?

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Context: Let us name the corner points of this symbol as shown.

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Context: Suppose the lengths of the arms of the symbol are $\text{AB} = 4\text{ cm}$ and $\text{BC} = 8\text{ cm}$ . Several such symbols can be constructed with the same lengths.

Q. To get the exact replica, would it help to take any other measurement?

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What do you think they can do?

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Q. Do you agree with Meera?

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Q10

Context: Sidelengths of a triangle are $4\text{ cm}$ , $6\text{ cm}$ , and $8\text{ cm}$ .

Q. Is this information sufficient to replicate the triangle with the same size and shape? If yes, can you do so?

Q11

Examine whether $\Delta \text{ABE}$ and $\Delta \text{ABF}$ are congruent.

Q12

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Q13

Are there other ways of overlapping the vertices so that the triangles fit exactly over each other?

Q14

Can you identify a pair of congruent triangles below? Why are they congruent?

Consider $\Delta ABD$ and $\Delta CDB$ . Since $ABCD$ is a rectangle, we have

\begin{aligned} \text{AB} &= \text{CD} \\ \text{AD} &= \text{CB} \end{aligned}

If the remaining sides of $\Delta ABD$ and $\Delta CDB$ have the same length then the SSS condition is satisfied, confirming the congruence of the two triangles. Is this the case?

Q15

Verify this by superimposing paper cutouts of the triangles obtained from the rectangle $ABCD$ (Fig. 1.1).

Q16

Identify the correct correspondence of vertices and express the congruence between the two triangles.

Q17

Suppose the angles are $30^\circ$ , $70^\circ$ , and $80^\circ$ . Can we create an exact copy of the frame with this?

Q18

$\Delta ABC$ and $\Delta XYZ$ are two triangles such that

$AB = XY = 6\text{ cm}$ , $AC = XZ = 5\text{ cm}$ , and $\angle A = \angle X = 30^\circ$

Are they congruent?

Q19

Context: $\Delta ABC$ and $\Delta XYZ$ are two triangles such that $AB = XY = 6\text{ cm}$ , $AC = XZ = 5\text{ cm}$ , and $\angle A = \angle X = 30^\circ$ .

Q. Construct a triangle having the above measurements.

Compare it with the triangles constructed by your classmates. Are the triangles all congruent? Explain why all such triangles with these measurements are congruent.

Q20

$\Delta ABC$ and $\Delta XYZ$ are two triangles such that

$AB = XY = 6\text{ cm}$ , $AC = XZ = 4\text{ cm}$ , and $\angle B = \angle Y = 30^\circ$

Are they congruent?

Q21

Context: $\Delta ABC$ and $\Delta XYZ$ are two triangles such that $AB = XY = 6\text{ cm}$ , $AC = XZ = 4\text{ cm}$ , and $\angle B = \angle Y = 30^\circ$ .

Q. Can there exist non-congruent triangles having these measurements? Construct and find out.

Q22

$\Delta \text{ABC}$ and $\Delta \text{XYZ}$ are two triangles with,

$\text{BC} = \text{YZ} = 5 \text{ cm}, \quad \angle \text{B} = \angle \text{Y} = 50^\circ \quad \text{and} \quad \angle \text{C} = \angle \text{Z} = 30^\circ$

Are they congruent?

Q23

Context: $\Delta \text{ABC}$ and $\Delta \text{XYZ}$ are two triangles with $\text{BC} = \text{YZ} = 5 \text{ cm}$ , $\angle \text{B} = \angle \text{Y} = 50^\circ$ , and $\angle \text{C} = \angle \text{Z} = 30^\circ$ .

Q. Can there exist non-congruent triangles having these measurements? Construct and find out.

Q24

In the figure, Point $O$ is the midpoint of $AD$ and $BC$ . What can one say about the lengths $AB$ and $CD$ ?

Q25

The following triangles $\Delta ABC$ and $\Delta XYZ$ are such that $\angle A = \angle X = 35^\circ$ , $\angle C = \angle Z = 75^\circ$ , and $BC = YZ = 4\text{ cm}$ . Are the triangles congruent? Give a reason.

Q26

$\Delta \text{ABC}$ and $\Delta \text{XYZ}$ are right-angled triangles such that $\text{BC} = \text{YZ} = 4\text{ cm}$ , $\angle \text{B} = \angle \text{Y} = 90^\circ$ and $\text{AC} = \text{XZ} = 5\text{ cm}$ . Are they congruent?

Q27

Context: $\Delta ABC$ and $\Delta XYZ$ are right-angled triangles such that $BC = YZ = 4\text{ cm}$ , $\angle B = \angle Y = 90^\circ$ , and $AC = XZ = 5\text{ cm}$ .

Q. Can there exist non-congruent triangles having these measurements? Construct and find out.

Q28

Consider the downward extension of line $l$ below $\text{QR}$ . Would the arc from $\text{R}$ meet this line downwards as well (as in the case of triangle construction when the sidelengths are given)? If so, would this lead to a triangle whose size and shape are different from $\Delta \text{PQR}$ , and yet has the given measurements?

Q29

$\Delta ABC$ is isosceles with $AB = AC$ , and $\angle A = 80^\circ$ . What can we say about $\angle B$ and $\angle C$ ?

Construct the altitude from $A$ to $BC$ .

Q30

Context: In $\Delta ABC$ , $AB = AC$ and $\angle A = 80^\circ$ .

Q. Can you use this fact to find $\angle B$ and $\angle C$ ?

Q31

Context: All the three angles of an equilateral triangle are equal.

Q. What could be their measures?

Q32

Context: In an equilateral triangle, each angle is $60^\circ$ .

Q. Verify this by construction.

Q33

Describe the congruent triangles you see in each picture.