Question

for circle q shown, ∠kqm is a central angle, and ∠kpm is a circumscribed angle.
which relationships are true?
select all the correct answers.
□ a. $overline{pm} \perp \overline{qm}$
□ b. $m\angle kpm + m\angle kqm = 180^\circ$
□ c. $m\angle kqm = \frac{1}{2}m\overarc{km}$
□ d. $m\angle kpm = \frac{1}{2}\left(m\overarc{klm} + m\overarc{km}\
ight)$
□ e. $overline{pk} \cong \overline{pm}$

Explanation:

Step1: Analyze Option A

A tangent to a circle is perpendicular to the radius at the point of tangency. \( \overline{PM} \) is a tangent (since \( \angle KPM \) is circumscribed, so \( PM \) and \( PK \) are tangents), and \( \overline{QM} \) is a radius. So \( \overline{PM} \perp \overline{QM} \). This is true.

Step2: Analyze Option B

Quadrilateral \( PKQM \) has two right angles (from tangents and radii: \( \angle PKQ = \angle PMQ = 90^\circ \)). The sum of interior angles of a quadrilateral is \( 360^\circ \), so \( m\angle KPM + m\angle KQM + 90^\circ + 90^\circ = 360^\circ \), which simplifies to \( m\angle KPM + m\angle KQM = 180^\circ \). This is true.

Step3: Analyze Option C

A central angle's measure is equal to the measure of its intercepted arc. So \( m\angle KQM = m\widehat{KM} \), not \( \frac{1}{2}m\widehat{KM} \). This is false.

Step4: Analyze Option D

The measure of a circumscribed angle (formed by two tangents) is \( \frac{1}{2} \) the difference of the measures of the intercepted arcs. The total circle is \( 360^\circ \), so \( m\widehat{KLM} = 360^\circ - m\widehat{KM} \). Then \( m\angle KPM=\frac{1}{2}(m\widehat{KLM} - m\widehat{KM}) \), not the sum. This is false.

Step5: Analyze Option E

Tangents from a common external point to a circle are congruent. \( P \) is an external point, so \( \overline{PK} \cong \overline{PM} \). This is true.

Answer:

A. \( \overline{PM} \perp \overline{QM} \)
B. \( m\angle KPM + m\angle KQM = 180^\circ \)
E. \( \overline{PK} \cong \overline{PM} \)