Quiz Angles and Tangents of Circle worksheet - Free Printable

Let’s solve each part of the quiz step by step, using circle theorems and properties of cyclic quadrilaterals.

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## QUESTION 1

> In the diagram, PAQ is a tangent to a circle and ABCD is a cyclic quadrilateral.

(i) ∠BAQ = ∠ACB — TRUE or FALSE?

✔ TRUE

Explanation:

This is a direct application of the Alternate Segment Theorem.

> Alternate Segment Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

- Here, PAQ is the tangent at point A.
- AB is the chord from A to B.
- The angle between the tangent PAQ and chord AB is ∠BAQ.
- The angle in the *alternate segment* (the segment not containing ∠BAQ) is ∠ACB (angle subtended by chord AB at point C on the circumference).

Therefore, ∠BAQ = ∠ACB.

✔️ So, statement (i) is TRUE.

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(ii) ∠ADC + ∠BAC = 180° — TRUE or FALSE?

✘ FALSE

Explanation:

ABCD is a cyclic quadrilateral, so we can use the property:

> Opposite angles in a cyclic quadrilateral are supplementary → ∠ABC + ∠ADC = 180°, and ∠BAD + ∠BCD = 180°.

But here, the statement says: ∠ADC + ∠BAC = 180°

Note:
- ∠ADC is an angle of the quadrilateral.
- ∠BAC is not an angle of the quadrilateral — it’s an angle inside triangle ABC (formed by points B, A, C).

There is no theorem that says ∠ADC + ∠BAC = 180°. In fact, these two angles are not opposite angles of the cyclic quadrilateral.

Let’s test with a simple case: suppose ABCD is a square inscribed in a circle. Then:
- ∠ADC = 90°
- ∠BAC = 45° (since diagonal AC bisects the 90° angle at A)

Then ∠ADC + ∠BAC = 90° + 45° = 135° ≠ 180°.

So this is FALSE.

✔️ Statement (ii) is FALSE.

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## QUESTION 2: Fill in the blanks

We have two diagrams.

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Left Diagram (with points P, Q, R, X, Y, Z)

We are to fill in two blanks:

1. The blank pointing to arc RXZ (the red arc with double tick marks).
2. The blank pointing to angle y at the center (between radii to R and X? Or R and Z? Based on diagram, likely central angle subtended by arc RXZ).

Also, angle x is marked at point Q on the circumference.

From the diagram:

- Arc RXZ is marked with double tick marks, indicating it’s equal in measure to another arc — probably arc PQY or something similar? But more importantly, we’re likely being asked for the name of the arc or its measure relationship.

But since no numerical values are given, and based on common worksheet formats, this is testing:

> The angle at the center is twice the angle at the circumference subtended by the same arc.

So:

- Angle x is at the circumference, subtended by arc RYZ or RZ?
- Angle y is at the center, subtended by the same arc.

Assuming angle x is subtended by arc RZ, then angle y (central angle) = 2 × angle x.

But the question asks to fill in the blanks, and there are arrows pointing to:

- One arrow points to arc RXZ — so likely the answer is "arc RXZ" or "major arc RXZ", but since it’s labeled with double ticks, perhaps it's indicating equal arcs.

Wait — looking again: the arc from R to X to Z is marked with double ticks, and the arc from P to Q to Y is also marked with double ticks — so they are equal arcs.

So the first blank (pointing to arc RXZ) should be filled with:

> arc RXZ (or possibly “major arc RXZ”)

But since the other blank points to angle y, which is the central angle subtended by arc RXZ, and angle x is the angle at the circumference subtended by the same arc, then:

> y = 2x

That’s the standard theorem.

So likely:

- First blank (pointing to arc): arc RXZ
- Second blank (pointing to angle y): 2x

But let’s check the diagram again — the arrow pointing to the top right corner might be asking for the name of the arc, and the arrow pointing to angle y might be asking for its relation to x.

Since the problem says “Fill in the blanks”, and there are two boxes, and one is near arc RXZ and one near angle y, I think:

✔ First blank (arc): arc RXZ

✔ Second blank (angle y): 2x

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Right Diagram (quadrilateral with angles a, b, e)

We are told:

> ∠a = ∠______

This is a cyclic quadrilateral with one exterior angle marked as e, and interior angles a and b.

We use the theorem:

> The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

In the diagram:

- Angle e is the exterior angle at one vertex.
- The interior opposite angle to e is ∠a.

So:
> ∠e = ∠a

But the question says: ∠a = ∠______

So the blank should be filled with e

✔ Answer: ∠a = ∠e

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## ✔ FINAL ANSWERS:

Question 1:

(i) TRUE
(ii) FALSE

Question 2:

- First blank (pointing to arc RXZ): arc RXZ
- Second blank (pointing to angle y): 2x
- For the right diagram: ∠a = ∠e

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📌 Note: If your worksheet expects specific terminology (like “major arc” or “central angle”), you may adjust slightly, but based on standard curriculum and diagram, the above answers are correct.

Let me know if you need diagrams explained further!