What are congruent angles? Justify with an example that in all the right-angled triangles, at least one pair of angles will be congruent. [2 MARKS]

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Solution

Definition: 1 Mark
Proof: 1 Mark

If two angles have the same measurement, they are congruent. Also, if two angles are congruent, their measurements are same.

We know that in every right-angled triangle, one angle is $90^{\circ}$ .

Example: Consider two right-angled triangles $Δ A J U$ and $Δ N I V$

In $Δ A J U, ∠ A J U = 90^{\circ}$

and in $Δ N I V, ∠ N I V = 90^{\circ}$

Since one angle is same in both, we can say that they have one pair of congruent angles.

So, if you take any two right-angled triangles, at least one pair of angles will be congruent, i.e. equal.

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Similar questions

Q. In each example below, the given triangles are congruent in the given correspondence. Write their congruent sides and congruent angles.

Q. In two right triangles one pair of acute angles and the sides opposite to them are equal. Prove that the triangles are congruent.

Question 7
Draw a $Δ A B C$ in which $A B$ = $4$ cm, $B C$ = $6$ cm and $A C$ = $9$ cm, construct a triangle similar to $Δ A B C$ with scale factor $\frac{3}{2}$ . Justify the construction. Are the two triangles congruent? Note that, all the three angles and two sides of the two triangles are equal.

Solution

Thinking process

Triangles are congruent when all corresponding sides and interior angles are congruent The triangles will have the same shape and size, but one may be a mirror image of the other.

So, first, we construct a triangle similar to $Δ A B C$ with scale factor $\frac{3}{2}$ and use the above concept to check the triangles are congruent or not.

Question 7
Draw a $Δ A B C$ in which AB= 4 cm, BC = 6 cm and AC = 9 cm, construct a triangle similar to $Δ A B C$ with scale factor $\frac{3}{2}$ . Justify the construction. Are the two triangles congruent? Note that, all the three angles and two sides of the two triangles are equal.

Thinking process

Triangles are congruent when all corresponding sides and interior angles are congruent The triangles will have the same shape and size, but one may be a mirror image of the other.

So, first, we construct a triangle similar to

Δ A B C

with scale factor

\frac{3}{2}

and use the above concept to check the triangles are congruent or not.

Q. AAS (or SAA) test :
The sum of the measures of angles in a triangle is

180^{\circ}

. Therefore, if two corresponding pairs of angles in two triangles are congruent, then the remaining pair of angles is also congruent. Hence, if two angles and a side adjacent to one of them are congruent with corresponding parts of the other triangle then the condition for ASA test is fulfilled. So the triangles are congruent.

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