If in two right triangles,one of the acute angles of one triangle is equal to an acute angle of the other triangle,can you say that the two triangles will be similar? Why?
(A) Yes,the two triangles will be similar.
Let the two right-angled triangles be $\triangle ABC$ and $\triangle PQR$,where $\angle B = \angle Q = 90^{\circ}$.
Given that one acute angle of the first triangle is equal to an acute angle of the second triangle,let $\angle A = \angle P$.
In $\triangle ABC$ and $\triangle PQR$:
$1$. $\angle B = \angle Q = 90^{\circ}$ (Given)
$2$. $\angle A = \angle P$ (Given)
By the $AA$ (Angle-Angle) similarity criterion,since two angles of one triangle are equal to two angles of the other triangle,the triangles are similar $(\triangle ABC \sim \triangle PQR)$.
This is also a specific case of the $AAA$ similarity criterion,as the third angles must also be equal due to the angle sum property of a triangle.
Let the two right-angled triangles be $\triangle ABC$ and $\triangle PQR$,where $\angle B = \angle Q = 90^{\circ}$.
Given that one acute angle of the first triangle is equal to an acute angle of the second triangle,let $\angle A = \angle P$.
In $\triangle ABC$ and $\triangle PQR$:
$1$. $\angle B = \angle Q = 90^{\circ}$ (Given)
$2$. $\angle A = \angle P$ (Given)
By the $AA$ (Angle-Angle) similarity criterion,since two angles of one triangle are equal to two angles of the other triangle,the triangles are similar $(\triangle ABC \sim \triangle PQR)$.
This is also a specific case of the $AAA$ similarity criterion,as the third angles must also be equal due to the angle sum property of a triangle.
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