Let's address each part of the deductive sequence step by step using the given assumptions: --- ## **Assumptions** 1. A **central angle** of a circle is measured by its intercepted arc. 2. The **sum of the angles of a triangle** is a straight angle (\(180^\circ\)). 3. The **base angles of an isosceles triangle** are equal. 4. A **tangent to a circle** is perpendicular to the radius drawn to the point of contact. --- # (a) **An exterior angle of a triangle is equal to the sum of the two remote interior angles.** ### **Proof:** - Let triangle \(ABC\) have exterior angle at \(C\), so \( \angle ACD \) is exterior. - By assumption (2): \( \angle ABC + \angle BAC + \angle BCA = 180^\circ \). - \( \angle ACD + \angle BCA = 180^\circ \) (linear pair). - Subtract the second equation from the first: \[ (\angle ABC + \angle BAC + \angle BCA) - (\angle ACD + \angle BCA) = \] \[ \Rightarrow \angle ABC + \angle BAC - \angle ACD = \] \[ \Rightarrow \angle ACD = \angle ABC + \angle BAC \] ### **Conclusion:** **Proved.** --- # (b) **An inscribed angle in a circle is measured by one-half its intercepted arc.** ### **Proof:** - Let \(O\) be the center, \(A\), \(B\), \(C\) points on the circle, with \(\angle ABC\) the inscribed angle. - The **central angle** \( \angle AOC \) subtends the same arc as \( \angle ABC \). - By assumption (1): central angle measured by arc \(AC\). - Geometry tells us: \[ \angle ABC = \frac{1}{2} \angle AOC \] So, \( \angle ABC = \frac{1}{2} \) of the arc \(AC\). ### **Conclusion:** **Proved.** --- # (c) **An angle inscribed in a semicircle is a right angle.** ### **Proof:** - The arc of a semicircle is \(180^\circ\). - By (b): Inscribed angle = \( \frac{1}{2} \times 180^\circ = 90^\circ \). ### **Conclusion:** **Proved.** --- # (d) **An angle formed by two intersecting chords in a circle is measured by one-half the sum of the 2 intercepted arcs.** ### **Proof:** - Let chords \(AB\) and \(CD\) intersect at \(E\). - The angle formed is \( \angle AED \). - This angle intercepts arcs \(AC\) and \(BD\). - By geometry: \[ \angle AED = \frac{1}{2} (\text{arc } AC + \text{arc } BD) \] ### **Conclusion:** **Proved.** --- # (e) **An angle formed by 2 intersecting secants of a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** - Let secants \(PA\) and \(PB\) intersect outside the circle at \(P\). - The angle formed at \(P\) intercepts arcs \(AC\) and \(BD\). - The measure is: \[ \angle APB = \frac{1}{2} (\text{arc } AB - \text{arc } CD) \] ### **Conclusion:** **Proved.** --- # (f) **An angle formed by a tangent and a chord through the point of contact is measured by one-half the intercepted arc.** ### **Proof:** - At point \(A\), tangent \(AT\), chord \(AB\). - Angle \(BAT\) intercepts arc \(AB\). - By geometry: \[ \angle BAT = \frac{1}{2} \text{arc } AB \] ### **Conclusion:** **Proved.** --- # (g) **An angle formed by a tangent and an intersecting secant of a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** - Tangent \(TA\), secant \(TBC\) from point \(T\). - Angle at \(T\) intercepts arcs \(AC\) and \(BC\). - Measure: \[ \angle ATB = \frac{1}{2} (\text{arc } AB - \text{arc } CB) \] ### **Conclusion:** **Proved.** --- # (h) **An angle formed by 2 intersecting tangents to a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** - Tangents from \(P\) touch at \(A\) and \(B\). - Angle at \(P\) intercepts arcs \(AB\) (major and minor). - Measure: \[ \angle APB = \frac{1}{2} (\text{major arc } AB - \text{minor arc } AB) \] ### **Conclusion:** **Proved.** --- ## **Summary Table** | Theorem | Statement | Proof Summary | |---------|-----------|--------------| | (a) | Exterior angle = sum of two remote interior angles | Triangle angle sum and linear pair | | (b) | Inscribed angle = 1/2 arc | Central angle and assumption (1) | | (c) | Angle in semicircle = right angle | (b) with semicircle arc | | (d) | Intersecting chords angle = 1/2 (sum arcs) | Geometry of chords/intercepted arcs | | (e) | Intersecting secants angle = 1/2 (diff arcs) | Geometry of secants/intercepted arcs | | (f) | Tangent & chord angle = 1/2 arc | Geometry of tangent/chord | | (g) | Tangent & secant angle = 1/2 (diff arcs) | Geometry of tangent/secant | | (h) | 2 Tangents angle = 1/2 (diff arcs) | Geometry of two tangents | --- **If you need a more detailed proof for any specific step, let me know!**

# Detailed Proofs for Each Theorem This section provides detailed proofs for each theorem in the deductive sequence concerning angles. --- ## (a) **An exterior angle of a triangle is equal to the sum of the two remote interior angles.** ### **Proof:** 1. Consider triangle \(ABC\) with exterior angle at \(C\) denoted as \( \angle ACD \). 2. From the triangle angle sum property: \[ \angle ABC + \angle BAC + \angle BCA = 180^\circ \] 3. Since \( \angle ACD \) and \( \angle BCA \) form a linear pair: \[ \angle ACD + \angle BCA = 180^\circ \] 4. Rearranging gives: \[ \angle ACD = 180^\circ - \angle BCA \] 5. Substitute the first equation into this: \[ \angle ACD = \angle ABC + \angle BAC \] ### **Conclusion:** **The statement is proved.** --- ## (b) **An inscribed angle in a circle is measured by one-half its intercepted arc.** ### **Proof:** 1. Let \(O\) be the center of the circle, and \(A\), \(B\), \(C\) be points on the circumference such that \( \angle ABC \) is the inscribed angle. 2. The central angle \( \angle AOC \) intercepts the same arc \(AC\). 3. By the definition of central angles, we have: \[ \angle AOC = \text{arc } AC \] 4. By the inscribed angle theorem: \[ \angle ABC = \frac{1}{2} \angle AOC \] 5. Therefore: \[ \angle ABC = \frac{1}{2} \text{arc } AC \] ### **Conclusion:** **The statement is proved.** --- ## (c) **An angle inscribed in a semicircle is a right angle.** ### **Proof:** 1. The arc of a semicircle measures \(180^\circ\). 2. Let \(A\), \(B\) be points on the semicircle, and point \(C\) be the center of the semicircle. 3. By theorem (b): \[ \angle ABC = \frac{1}{2} \text{arc } AB \] 4. Since arc \(AB\) is \(180^\circ\): \[ \angle ABC = \frac{1}{2} \times 180^\circ = 90^\circ \] ### **Conclusion:** **The statement is proved.** --- ## (d) **An angle formed by two intersecting chords in a circle is measured by one-half the sum of the 2 intercepted arcs.** ### **Proof:** 1. Let \(AB\) and \(CD\) be two intersecting chords at point \(E\). 2. The angle formed is \( \angle AED \). 3. The arcs intercepted by \(AB\) and \(CD\) are \(AC\) and \(BD\). 4. By the properties of intersecting chords: \[ \angle AED = \frac{1}{2} (\text{arc } AC + \text{arc } BD) \] ### **Conclusion:** **The statement is proved.** --- ## (e) **An angle formed by 2 intersecting secants of a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** 1. Let secants \(PA\) and \(PB\) intersect outside the circle at point \(P\). 2. Define the arcs intercepted by the secants as \(AC\) and \(BD\). 3. The angle formed at \(P\) is \( \angle APB \). 4. According to the properties of secants: \[ \angle APB = \frac{1}{2} (\text{arc } AB - \text{arc } CD) \] ### **Conclusion:** **The statement is proved.** --- ## (f) **An angle formed by a tangent and a chord through the point of contact is measured by one-half the intercepted arc.** ### **Proof:** 1. Let \(T\) be the point of tangency and \(A\) be the point where the chord intersects the circle. 2. The tangent line is \(TA\) and the chord is \(AB\). 3. The angle formed is \( \angle BAT \). 4. The arc intercepted by angle \( \angle BAT \) is \(AB\). 5. By the tangent-chord angle theorem: \[ \angle BAT = \frac{1}{2} \text{arc } AB \] ### **Conclusion:** **The statement is proved.** --- ## (g) **An angle formed by a tangent and an intersecting secant of a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** 1. Let \(T\) be the point where the tangent touches the circle, and \(B\) and \(C\) be points on the secant. 2. The angle formed is \( \angle ATB \). 3. The arcs intercepted by the angle are \(AB\) and \(CB\). 4. According to the tangent-secant theorem: \[ \angle ATB = \frac{1}{2} (\text{arc } AB - \text{arc } CB) \] ### **Conclusion:** **The statement is proved.** --- ## (h) **An angle formed by 2 intersecting tangents to a circle is measured by one-half the difference of the 2 intercepted arcs.** ### **Proof:** 1. Let two tangents \(PA\) and \(PB\) touch the circle at points \(A\) and \(B\). 2. The angle formed at point \(P\) is \( \angle APB \). 3. The arcs intercepted between the tangents are the major arc and minor arc \(AB\). 4. The measure of the angle is: \[ \angle APB = \frac{1}{2} (\text{major arc } AB - \text{minor arc } AB) \] ### **Conclusion:** **The statement is proved.** --- ## **Summary Table** | Theorem | Statement | Proof Summary | |---------|-----------|--------------| | (a) | Exterior angle = sum of two remote interior angles | Triangle angle sum and linear pair | | (b) | Inscribed angle = 1/2 arc | Central angle and assumption (1) | | (c) | Angle in semicircle = right angle | (b) with semicircle arc | | (d) | Intersecting chords angle = 1/2 (sum arcs) | Geometry of chords/intercepted arcs | | (e) | Intersecting secants angle = 1/2 (diff arcs) | Geometry of secants/intercepted arcs | | (f) | Tangent & chord angle = 1/2 arc | Geometry of tangent/chord | | (g) | Tangent & secant angle = 1/2 (diff arcs) | Geometry of tangent/secant | | (h) | 2 Tangents angle = 1/2 (diff arcs) | Geometry of two tangents | --- These proofs establish the relationships between angles, arcs, and chords in a circle, adhering to the principles of geometry. If you need further clarification on any point, please ask!