Question

use the given information to prove that ∠5 and ∠8 are right angles. given: ∠6≅∠8, lines j and k are perpendicular. prove: ∠5≅∠8. statement: 1. lines j and k are perpendicular. 2. ∠□ and ∠□ are right angles. 3. ∠5≅∠□. 4. ∠6≅∠8. 5. ∠5≅∠8. reason: 1. given. 2. select. 3. select. 4. given. 5. select.

Explanation:

Step1: Recall perpendicular - line property

If two lines \(j\) and \(k\) are perpendicular, then the angles formed by them are right - angles. Let the intersection of \(j\) and \(k\) form angles. Since \(j\perp k\), the adjacent angles formed are right - angles. Let's assume \(\angle5\) and \(\angle6\) are adjacent angles formed by \(j\) and \(k\).

Step2: Use the given congruence

We are given that \(\angle6\cong\angle8\). Also, since \(j\perp k\), \(\angle5\) and \(\angle6\) are right - angles (by the definition of perpendicular lines, which form 90 - degree angles).

Step3: Apply the transitive property of congruence

Since \(\angle6\cong\angle8\) and \(\angle5\) and \(\angle6\) are right - angles, and congruent angles have equal measures, \(\angle5\cong\angle8\) and both \(\angle5\) and \(\angle8\) are right - angles.

Answer:

In the statement column:

• For the third row (Statement 3): \(\angle5\cong\angle6\) (because when two lines are perpendicular, adjacent angles formed are congruent as they are both right - angles)
• For the fourth row (Statement 4): \(\angle6\cong\angle8\) (Given)
• For the fifth row (Statement 5): \(\angle5\cong\angle8\) (Transitive property of congruence)

In the reason column:

• For the third row (Reason 3): Definition of perpendicular lines (adjacent angles formed by perpendicular lines are right - angles and right - angles are congruent)
• For the fourth row (Reason 4): Given
• For the fifth row (Reason 5): Transitive property of congruence (If \(a = b\) and \(b = c\), then \(a = c\))