Question

given: ∠6≅∠8, lines j and k are perpendicular. prove: ∠5≅∠8. lines j and k are perpendicular. ∠4 and ∠ are right angles. ∠5≅∠. ∠6≅∠8. ∠5≅∠8. given. select. given. select. given. select.

Explanation:

Step1: Recall perpendicular - line property

If two lines are perpendicular, the angles formed at their intersection are right - angles. So, if lines $j$ and $k$ are perpendicular, $\angle4$ and $\angle 6$ are right angles (since they are the angles formed by the perpendicular lines $j$ and $k$).

Step2: Recall vertical - angle property

Vertical angles are congruent. $\angle5$ and $\angle7$ are vertical angles, and $\angle6$ and $\angle8$ are vertical angles. Given $\angle6\cong\angle8$. Also, $\angle5$ and $\angle7$ are vertical angles, and since $\angle4$ and $\angle6$ are right - angles, and considering the linear - pair and vertical - angle relationships, we know that $\angle5$ and $\angle8$ are congruent. The reason for $\angle5\cong\angle8$ is the Transitive Property of Congruence. Since $\angle6\cong\angle8$ (given) and $\angle5$ and $\angle7$ are related to the right - angles formed by the perpendicular lines and through vertical - angle and linear - pair relationships, we can prove $\angle5\cong\angle8$.
The missing angle in " $\angle5\cong\angle\square$ " is $\angle7$ (because vertical angles are congruent). For the reasons:

• For the statement " $\angle4$ and $\angle\square$ are right angles", the missing angle is $\angle6$ (because lines $j$ and $k$ are perpendicular, and the angles formed at their intersection are right - angles). The reason for " $\angle4$ and $\angle6$ are right angles" is "Definition of Perpendicular Lines".
• For " $\angle5\cong\angle7$", the reason is "Vertical Angles are Congruent".
• For " $\angle5\cong\angle8$", the reason is "Transitive Property of Congruence" (since $\angle6\cong\angle8$ and through the relationships of angles formed by perpendicular lines and vertical angles).

Answer:

• In the statement " $\angle4$ and $\angle\square$ are right angles", the answer is $\angle6$.
• In the statement " $\angle5\cong\angle\square$", the answer is $\angle7$.
• Reason for " $\angle4$ and $\angle6$ are right angles": Definition of Perpendicular Lines.
• Reason for " $\angle5\cong\angle7$": Vertical Angles are Congruent.
• Reason for " $\angle5\cong\angle8$": Transitive Property of Congruence.