Question

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Answer 1

m∠DFA = 115°

Explanation:

Given:

• m∡BC = 164°
• m∠DCF = 33°

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

⇒ m∠CEB = m∠CAB = m∠CDB = m∡BC ÷ 2

= 164° ÷ 2

= 82°

The interior angles of a triangle sum to 180°

⇒ m∠DCF + m∠CDF + m∠DFC = 180°

⇒ 33° + 82° + m∠DFC = 180°

⇒ m∠DFC = 65°

Angles on a straight line sum to 180°

⇒ m∠DFC + m∠DFA = 180°

⇒ 65° + m∠DFA = 180°

⇒ m∠DFA = 115°

Answer 2

115°

Explanation:

The measure of angle DFA where chords DB and CA cross is half the sum of arcs DA and CB. Arc CB is given as 164°, but arc DA must be figured from the inscribed angle that subtends it, angle DCA. That angle is given as 33°.

An arc is twice the measure of the inscribed angle it subtends, so arc DA is ...

arc DA = angle DCA × 2 = 33° × 2 = 66°

angle DFA = (66° +164°)/2 = 230°/2

angle DFA = 115°