Question

given: $overline{ig}perpoverline{ik}$, $overline{gm}perpoverline{km}$, $angle1congangle2$ prove: $\triangle gkicong\triangle kgm$ 1. given 2. perpendicular lines intersect to form right angles. 3. $angle icongangle m$

Explanation:

Step1: State given information

$\overline{IG}\perp\overline{IK},\overline{GM}\perp\overline{KM},\angle1\cong\angle2$

Step2: Find right - angle measures

Since $\overline{IG}\perp\overline{IK}$ and $\overline{GM}\perp\overline{KM}$, $\angle I = 90^{\circ}$ and $\angle M = 90^{\circ}$

Step3: Prove angle congruence

All right angles are congruent, so $\angle I\cong\angle M$

Step4: Identify common side

$\overline{GK}$ is common to both $\triangle GKI$ and $\triangle KGM$, so $\overline{GK}=\overline{GK}$ by reflexive property

Step5: Prove triangle congruence

By AAS (Angle - Angle - Side) congruence criterion with $\angle1\cong\angle2,\angle I\cong\angle M,\overline{GK}=\overline{GK}$, we have $\triangle GKI\cong\triangle KGM$

Answer:

1. $\overline{IG}\perp\overline{IK},\overline{GM}\perp\overline{KM},\angle1\cong\angle2$
2. $\angle I = 90^{\circ},\angle M=90^{\circ}$ (Perpendicular lines intersect to form right - angles)
3. $\angle I\cong\angle M$ (All right angles are congruent)
4. $\overline{GK}=\overline{GK}$ (Reflexive property of congruence)
5. $\triangle GKI\cong\triangle KGM$ (AAS congruence criterion since $\angle1\cong\angle2,\angle I\cong\angle M,\overline{GK}=\overline{GK}$)