Question

prove the following statement using the given pieces of information. given: ∠gks≅∠jks, ks⊥gj. prove: △sgk≅△sjk. complete the two - column proof. statements: 1. ks⊥gj; 2. ∠ksg and ∠ksj are right angles; 3. ∠ksg≅∠ksj

Explanation:

Step1: Define right - angles

Given \(KS\perp GJ\), by the definition of perpendicular lines, \(\angle KSG\) and \(\angle KSJ\) are right angles.

Step2: Apply right - angles congruence

Since \(\angle KSG\) and \(\angle KSJ\) are right angles, by the Right Angles Congruent Theorem, \(\angle KSG\cong\angle KSJ\).

Step3: Identify given angle

We are given that \(\angle GKS\cong\angle JKS\).

Step4: Identify common side

Side \(KS = KS\) by the Reflexive Property of Congruence.

Step5: Prove triangle congruence

By the Angle - Side - Angle (ASA) Congruence Postulate, since \(\angle GKS\cong\angle JKS\), \(KS\cong KS\), and \(\angle KSG\cong\angle KSJ\), we have \(\triangle SGK\cong\triangle SJK\).

Answer:

\(\triangle SGK\cong\triangle SJK\)