The situation in which triangles will be made congruent for sure is given by: Option ( iii ) Both making a triangle TRY with right angle at T and sides TR =4 cm and RY =5cm
What are congruent triangles?
Suppose it is given that two triangles ΔABC ≅ ΔDEF
Then that means ΔABC and ΔDEF are congruent. Congruent triangles are exact same triangles, but they might be placed at different positions.
The order in which the congruency is written matters.
For ΔABC ≅ ΔDEF, we have all of their corresponding elements like angle and sides congruent.
Thus, we get:
[tex]\rm m\angle A = m\angle D \: or \: \: \angle A \cong \angle D \angle B = \angle E\\\\\rm m\angle B = m\angle E \: or \: \: \angle B \cong \angle E \\\\\rm m\angle C = m\angle F \: or \: \: \angle C \cong \angle F \\\\\rm |AB| = |DE| \: \: or \: \: AB \cong DE\\\\\rm |AC| = |DF| \: \: or \: \: AC \cong DF\\\\\rm |BC| = |EF| \: \: or \: \: BC \cong EF[/tex]
(|AB| denotes length of line segment AB, and so on for others).
For two triangles being congruent, we need at least:
- Either, 3 corresponding side pairs are congruent. (SSS congruency),
- Or, any 2 corresponding sides and angle between them are congruent. (SAS congruency).
- Or, any 1 side of same measurement as its corresponding side in other triangle with two angles on its endpoints of same measurement as per their corresponding angles in the other triangle. (ASA congruency).
Checking all the options:
- ( i ) Both making a triangle FUN with side FU=4cm and angle F= 60º
Only one side and one angle are not enough.
- ( ii ) Both making an equilateral triangle ARE.
Two equilateral triangles are not necessary to be congruent. Example, one equilateral triangle has sides of 3 units, and other triangle has sides of 5 units.
- ( iii ) Both making a triangle TRY with right angle at T and sides TR =4 cm and RY =5cm
Two sides TR and RY and a non-included angle T is given. This doesn't satisfy SAS' condition. But since it is given that:
Angle T is right angled, so we now know that RY is going to be hypotenuse.
Using this fact, and the fact that |RY| = 5 cm, |TR| = 4 cm, and Pythagoras theorem, we can get |TY| = 3 cm.
Thus, it will complete SSS condition, and therefore, by SSS congruency, the triangles made in this case would be congruent.
- ( iv ) Both making triangle DIP with all angles equal.
All angles(corresponding) being equal implies similarity, but not congruence.
Thus, the situation in which triangles will be made congruent for sure is given by: Option ( iii ) Both making a triangle TRY with right angle at T and sides TR =4 cm and RY =5cm
Learn more about congruent triangles here:
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