# Lesson 4

### Problem 1

A quadrilateral $$ABCD$$ has the given angle measures. Select all measurements which could come from a cyclic quadrilateral.

A:

angle $$A$$ is 90$$^\circ$$, angle $$B$$ is 90$$^\circ$$, angle $$C$$ is 90$$^\circ$$, and angle $$D$$ is 90$$^\circ$$

B:

angle $$A$$ is 80$$^\circ$$, angle $$B$$ is 80$$^\circ$$, angle $$C$$ is 100$$^\circ$$, and angle $$D$$ is 100$$^\circ$$

C:

angle $$A$$ is 70$$^\circ$$, angle $$B$$ is 110$$^\circ$$, angle $$C$$ is 70$$^\circ$$, and angle $$D$$ is 110$$^\circ$$

D:

angle $$A$$ is 60$$^\circ$$, angle $$B$$ is 50$$^\circ$$, angle $$C$$ is 120$$^\circ$$, and angle $$D$$ is 130$$^\circ$$

E:

angle $$A$$ is 50$$^\circ$$, angle $$B$$ is 40$$^\circ$$, angle $$C$$ is 120$$^\circ$$, and angle $$D$$ is 150$$^\circ$$

### Problem 2

Quadrilateral $$ABCD$$ is cyclic with given angle measures.

1. What is the measure of angle $$C$$?
2. What is the measure of angle $$D$$?

### Problem 3

Lin is looking at cyclic quadrilateral $$ABCD$$. She says, “I’m not convinced that opposite angles of cyclic quadrilaterals always add up to 180 degrees. For example, in this diagram, suppose we moved point $$A$$ to a different spot on the circle. Angle $$BCD$$ would still measure 100 degrees, but now angle $$BAD$$ would have a different measure, and they wouldn’t add up to 180.”

Do you agree with Lin? Explain or show your reasoning.

### Problem 4

Line $$AC$$ is tangent to the circle centered at $$O$$ with radius 3 units. The length of segment $$AC$$ is 4.5 units. Find the length of segment $$AB$$.

A:

$$3+\sqrt{29.25}$$ units

B:

$$\sqrt{29.25}$$ units

C:

$$\text-3+\sqrt{29.25}$$ units

D:

26.25 units

### Solution

(From Unit 7, Lesson 3.)

### Problem 5

Technology required. Line $$PD$$ is tangent to a circle of radius 1 inch centered at $$O$$. The length of segment $$PD$$ is 1.2 inches. The length of segment $$AB$$ is 1.7 inches. Han is trying to figure out if $$C$$ or $$B$$ is closer to $$P$$. He uses the Pythagorean Theorem to find the length of $$OP$$. Then he subtracts 1 from the length of $$OP$$ to determine how far $$C$$ is from point $$P$$.

1. How far is $$B$$ from point $$P$$?
2. Which point is closest to $$P$$? Explain your reasoning.

### Solution

(From Unit 7, Lesson 3.)

### Problem 6

In the diagram, the measure of angle $$ACB$$ is 25 degrees. What is the measure of angle $$AOB$$?

### Solution

(From Unit 7, Lesson 2.)

### Problem 7

Which statement must be true?

A:

A diameter is a chord.

B:

C:

A chord is a diameter.

D:

A central angle’s vertex is on the circle.

### Solution

(From Unit 7, Lesson 1.)

### Problem 8

A circle and line are drawn. How many intersection points are possible? Select all possible answers.

A:

0

B:

1

C:

2

D:

3

E:

4