Question

tu ⊥ st and pq ⊥ qr. complete the proof that ∠pqr ≅ ∠stu.

	statement	reason
1	tu ⊥ st	given
2	pq ⊥ qr	given
3	m∠stu = 90°
4	m∠pqr = 90°	definition of perpendicular lines
5	m∠stu = m∠pqr	transitive property of equality
6	∠pqr ≅ ∠stu

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not feeling ready yet? these can help:
learn about complementary angles!

• what are complementary angles?
• finding measures of complementary angles

lesson: complementary angles
lesson: vertical angles
properties of equality
lesson: supplementary angles
lesson: adjacent angles

Explanation:

Step1: Analyze Statement 4's Reason

Statement 4 is \( m\angle PQR = 90^\circ \). Since \( \overleftrightarrow{PQ} \perp \overleftrightarrow{QR} \) (given in Statement 2), the reason for \( m\angle PQR = 90^\circ \) is the "Definition of perpendicular lines" (because perpendicular lines form right angles, which measure \( 90^\circ \)).

Step2: Analyze Statement 6's Reason

Statement 5 is \( m\angle STU = m\angle PQR \) (by Transitive Property, since both equal \( 90^\circ \)). Statement 6 is \( \angle PQR \cong \angle STU \). The reason for this is the "Definition of congruent angles" (if two angles have equal measures, they are congruent).

Answer:

• For Statement 4: Reason is "Definition of perpendicular lines"
• For Statement 6: Reason is "Definition of congruent angles"