The triangles Q and S are congruent according to the AAS congruence axiom. Both triangles have the same measure of two angles and one non-included side.
What are congruent triangles?
Two triangles are said to be congruent if the three sides and the three angles of both the angles are equal in any orientation.
These triangles can be rotated, flipped, and turned to be looked identical. If repositioned, they coincide with each other. The symbol of congruence is’ ≅’.
What are the rules of congruency?
Rules of congruency are as follows:
SSS (Side-Side-Side): If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by the SSS rule.
SAS (Side-Angle-Side): If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.
ASA (Angle-Side-Angle): If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and the side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.
AAS (Angle-Angle-Side): When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.
RHS (Right angle-Hypotenuse-Side): If the hypotenuse and a side of a right-angled triangle are equivalent to the hypotenuse and a side of the second right-angled triangle, then the two right triangles are said to be congruent by the RHS rule.
Finding the congruency of the given triangles:
The given triangles are ΔQ, ΔR, ΔS, and ΔT.
The ΔQ has dimensions as follows: (consider three vertices A, B, and C)
∠A = 65°, ∠C = 45° and non-included side AB = 10
So, the ∠B = 180° - 65° - 45° = 70°
The ΔR has dimensions as follows: (consider three vertices D, E, and F)
∠D = 45°, ∠E = 60° and non-included side EF = 10
So, the ∠F = 180° - 60° - 45° = 75°
The ΔS has dimensions as follows: (consider three vertices A', B', and C')
∠A' = 70°, ∠B' = 45° and non-included side A'C' = 10
So, the ∠C' = 180° - 70° - 45° = 65°
The ΔT has dimensions as follows: (consider three vertices D', E', and F')
∠D' = 65°, ∠E' = 45° and the included side D'E' = 10
So, the ∠F' = 180° - 65° - 45° = 70°
Since the ΔT has the included side as 10 and the ΔR has different angles on comparing with the other triangles, they are not able to become congruent with any other of the given triangles.
The remaining two triangles (ΔQ and ΔS) are similar in shape and have two angle measures and one non-included side length. So, they are congruent triangles according to the AAS congruency axiom.
I.e., ∠A = ∠B' = 65°, ∠C = ∠B' = 45° and non-included side AB = A'C' = 10.
So, the ΔQ and ΔS are congruent (Angle - Angle - Side).
∴ ΔQ ≅ ΔS (AAS - axiom)
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