NYS Geometry Regents Exam: Question 33 (January 2020)

NYS Geometry Regents Exam: Question 33 (January 2020)

This video provides a step-by-step solution to question 33 from the January 2020 NYS Geometry Regents exam. It's a valuable resource for students preparing for the exam, offering insights into common problem types and effective solution strategies.

**Question 33:**

In the diagram below, quadrilateral ABCD is inscribed in circle O. Diagonals AC and BD intersect at E.

If m∠ABC = 70 and m∠ADC = 110, what is the measure of ∠AED?

**Solution:**

1. **Inscribed Angle Theorem:** Recall that an inscribed angle's measure is half the measure of its intercepted arc. Therefore:

• m∠ABC = 1/2 * marc ADC
• m∠ADC = 1/2 * marc ABC

2. **Substitute and Solve:** Substitute the given angle measures into the equations:

• 70 = 1/2 * marc ADC => marc ADC = 140
• 110 = 1/2 * marc ABC => marc ABC = 220

3. **Central Angle Theorem:** A central angle's measure is equal to the measure of its intercepted arc. Therefore:

• m∠AOC = marc ADC = 140
• m∠BOD = marc ABC = 220

4. **Angle Addition Postulate:** The measure of a whole angle is equal to the sum of the measures of its parts:

• m∠AOB + m∠BOC + m∠COD + m∠DOA = 360

5. **Solve for ∠AOB:**

• m∠AOB + 140 + 220 + m∠DOA = 360
• m∠AOB + m∠DOA = 0

6. **Vertical Angles:** Vertical angles are congruent, meaning they have the same measure:

• m∠AOB = m∠DOA = 0

7. **Angle Addition Postulate (Again):**

• m∠AED = m∠AEO + m∠DEO
• m∠AED = 1/2 * m∠AOB + 1/2 * m∠DOA
• m∠AED = 1/2 * 0 + 1/2 * 0 = 0

Therefore, the measure of ∠AED is **0 degrees**. It's important to note that in this case, line segments AC and BD are actually diameters of circle O, making E the center of the circle.

**Key Takeaways:**

• This problem tests your knowledge of inscribed angles, central angles, and vertical angles.
• It's crucial to understand the relationships between angles and arcs in a circle.
• Remember that the sum of the angles in a quadrilateral is always 360 degrees.

By working through this problem step-by-step, you gain valuable practice in applying key geometry concepts and techniques, which can be helpful in preparing for the NYS Geometry Regents exam.