Given that ST = RT and PS and QR are produced to meet at T, the
congruency of the angles and the included side proves;
- ΔPRS ≅ ΔQRS by Angle-Side-Angle rule of congruency
How can the triangles be shown to be congruent?
The given parameters are;
The given cyclic quadrilateral = PQRS
The produced side that meet at point T = PS and QR
ST = RT
Required:
To prove that PRS = QRS
Solution:
A two column proof is presented as follows;
Statement [tex]{}[/tex] Reason
1. ST = RT [tex]{}[/tex] 1. Given
2. ΔSRT is an isosceles triangle [tex]{}[/tex] 2. Definition of isosceles triangle
3. ∠TSR ≅ ∠TRS [tex]{}[/tex] 3. Base angles of an isosceles triangle
4. ∠TSR = ∠TRS [tex]{}[/tex] 4. Definition of congruency
5. ∠TSR and ∠PSR, ∠TRS and ∠QRS are linear pair angles 5. Definition of linear pair angles
6. ∠TSR + ∠PSR = 180° and ∠TRS +∠QRS = 180° 6. Property of linear pair angles
7. ∠TSR + ∠PSR = ∠TRS +∠QRS, 7. Substitution property of equality
8. ∠PSR = ∠QRS [tex]{}[/tex] 8. Subtraction property of equality
9. ∠RQS in ΔRQS = ∠SPR in ΔSPR [tex]{}[/tex] 9. Angle subtending the same arc formed at the circumference of a circle
10. ΔPRS ≅ ΔQRS [tex]{}[/tex] 10. By ASA rule of congruency
The Angle-Side-Angle rule of congruency states that if two angles and
an included side of one triangle are congruent to two angles and the
included side of another triangle, then the two triangles are congruent.
Learn more about congruency here:
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