Question

PLEASE HELP :)
Complete the following proof.
Given:
AE tangent to circle R at A and to circle S at E
Prove:
∠R = ∠S

Answer 1

∠R in triangle ΔRCA and ∠S in triangle ΔCES are alternate interior angles.

Response;

• ∠R = ∠S by the alternate interior angles theorem

Method used to arrive at the proof;

Given;

The tangent to circle R at A and circle S at E = AE

Required:

Complete proof that ∠R = ∠S

Solution:

A two column proof is presented as follows;

Statement:            ${}$                                    Reasons:

RA and SE are radii of given circles        Definition

∠RAC in ΔRCA = 90°  ${}$                              Definition of tangent to a circle

∠CES in ΔCES = 90°  ${}$                               Tangent definition

∠ACR ≅ ∠ECS  ${}$                                         Vertical angles theorem

∠ACR = ∠ECS   ${}$                                         Definition of congruency

ΔRCA ~ ΔCES   ${}$                                         A-A similarity postulate

∠R ≅ ∠S   ${}$                                                  CASTC

∠R = ∠S   ${}$                                                   Definition of congruency

By second method, we have;

∠RAC = ∠CES = 90°   ${}$                               Definition of tangent to a circle

RA ║ ES    ${}$                             Converse of Alternate interior angles theorem

∠R ≅ ∠S    ${}$                                                  Alternate interior angles theorem

∠R = ∠S    ${}$   ��                                               Definition of congruency

• CASTC is the acronym for Corresponding Angles of Similar Triangles are Congruent.

• Angle-Angle, A-A, similarity postulate states that two triangles are similar if two angles in one triangle are congruent to two angles in the other triangle.

• The converse of the alternate interior angles theorem states that if the alternate interior angles formed between two lines and a common transversal are are congruent, the lines are parallel.

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