Question

Match the following reasons with the statement givem

Answer 1

Answer:

AAS(Angle-Angle-Side) postulate states that if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent

In triangle RAS and triangle QAT

$\angle R =\angle Q$   [Angle]

$AS =AT$    [Side]                                       [Given]

By Base Angle Theorem states that in an isosceles triangle(i.e, AST), the angles opposite the congruent sides(AS =AT) are congruent.

⇒ $\angle 5= \angle 6$      [By base ∠'s of isosceles triangle are equal]

By definition of supplementary angles, if two Angles are Supplementary when they add up to 180 degrees.

$\angle 4$, $\angle 5$ are supplementary and $\angle 6$, $\angle 7$ are supplementary.

⇒$\angle 4+ \angle 5 =180^{\circ}$ and

$\angle 6+ \angle 7 =180^{\circ}$  

Two $\angle 's$ supplementary to equal $\angle 's$

$\angle 4+ \angle 5 =\angle 6+ \angle 7$

Since,  $\angle 5 =\angle 6$  

then, we get;

$\angle 4 =\angle 7$  [Angle]

then, by AAS postulates,

$\triangle RAS \cong \triangle QAT$

By CPCT[Corresponding Part of Congruent Triangles are equal]

$RS = QT$              Hence Proved!