Answer:
[tex]\boxed{\sf Yes}\;;\textsf{because $\angle 9$ and $\angle7$ are \boxed{\sf vertical} angles, $m \angle 9=$ \boxed{\sf 90\; degrees}}\;.[/tex]
[tex]\textsf{Because $\angle6$ and $\angle8$ are \boxed{\sf complementary} angles, $m\angle6+m\angle8=$\;\boxed{\sf 90\;degrees}}\;.[/tex]
[tex]\textsf{Thus}, \; \boxed{\sf m\angle9 = m\angle6 + m\angle8}\;.[/tex]
Step-by-step explanation:
Vertical Angles Theorem
When two straight lines intersect, the opposite vertical angles are congruent.
Therefore, ∠9 and ∠7 are vertical angles and:
⇒ m∠9 = m∠7 = 90°
Angles on a straight line sum to 180°
⇒ m∠6 + m∠8 + m∠7 = 180°
⇒ m∠6 + m∠8 + 90° = 180°
⇒ m∠6 + m∠8 = 90°
Complementary Angles
Two angles whose measures sum to 90°.
Therefore, ∠6 and ∠8 are complementary angles and:
⇒ m∠6 + m∠8 = 90°
As m∠9 = 90° and m∠6 + m∠8 = 90° then:
⇒ m∠6 + m∠8 = m∠9
