1. **Theorem 1 (Example):** Given angles $\angle A$ and $\angle B$ such that $m\angle A = 30^\circ$ and $m\angle B = 45^\circ$. If $\angle A$ and $\angle B$ are complementary, show that their sum is $90^\circ$. Step 1: Recall the definition of complementary angles. Step 2: Calculate $m\angle A + m\angle B = 30^\circ + 45^\circ = 75^\circ$ which is less than $90^\circ$. Step 3: Since they sum to $75^\circ$, they are not complementary; for them to be complementary, one angle would need to be $60^\circ$ if the other is $30^\circ$. So Theorem 1: Complementary angles sum to $90^\circ$. 2. **Theorem 2 (Example):** If $\angle C$ and $\angle D$ are vertical angles and $m\angle C = 50^\circ$, find $m\angle D$. Step 1: Recall vertical angles are equal. Step 2: So $m\angle D = 50^\circ$. 3. **Theorem 3 (Example):** If $\triangle ABC$ is isosceles with $AB = AC$, show that $\angle B = \angle C$. Step 1: By definition, equal sides opposite equal angles. Step 2: Since $AB = AC$, then $m\angle B = m\angle C$. 4. **Theorem 4 (Example):** In any triangle, the sum of interior angles is $180^\circ$. Step 1: Consider $\triangle DEF$ with angles $D, E, F$. Step 2: Measure or assign $m\angle D = 40^\circ$, $m\angle E = 60^\circ$. Step 3: Calculate $m\angle F = 180^\circ - 40^\circ - 60^\circ = 80^\circ$. 5. **Arc Addition Postulate (Example):** Given arc $AB$ with points $A, C, B$ on circle, if $m\overset{\frown}{AC} = 40^\circ$ and $m\overset{\frown}{CB} = 50^\circ$, find $m\overset{\frown}{AB}$. Step 1: By arc addition postulate, $m\overset{\frown}{AB} = m\overset{\frown}{AC} + m\overset{\frown}{CB}$. Step 2: So $m\overset{\frown}{AB} = 40^\circ + 50^\circ = 90^\circ$. 6. **Sector of a Circle (Example):** Find the area of a sector with radius $r=10$ units and central angle $\theta=60^\circ$. Step 1: Area of sector is given by $\text{Area} = \pi r^2 \times \frac{\theta}{360}$. Step 2: Substitute values: $\pi \times 10^2 \times \frac{60}{360} = \pi \times 100 \times \frac{1}{6} = \frac{100\pi}{6} = \frac{50\pi}{3}$ units squared. 7. **Arc Length (Example):** Find arc length of circle with radius $r=7$ units and central angle $\theta=90^\circ$. Step 1: Arc length $L = 2\pi r \times \frac{\theta}{360}$. Step 2: Substitute values: $2\pi \times 7 \times \frac{90}{360} = 14\pi \times \frac{1}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}$ units. Final answers: - Theorem 1: $90^\circ$ (complementary angles). - Theorem 2: $50^\circ$ (vertical angles). - Theorem 3: Equal base angles. - Theorem 4: $180^\circ$ total. - Arc Addition Postulate: $90^\circ$ arc. - Sector area: $\frac{50\pi}{3}$ units squared. - Arc length: $\frac{7\pi}{2}$ units.