From the image, determine the geometric properties and calculate unknown angles in the given diagrams.

Solution:

The image contains several geometry problems involving circles and cyclic quadrilaterals. Let's analyze each problem:

Problem 1 (Top Left): Cyclic Quadrilateral ABCD

• Given: $$\angle DAC = 42^\circ$$, $$\angle ABD = 66^\circ$$.
• Find: $$\angle ABC = ?$$
• Analysis: In a cyclic quadrilateral, angles subtended by the same arc are equal. Also, opposite angles sum to 180 degrees.
• In $$\triangle ABD$$, $$\angle ADB = 180^\circ - 90^\circ - 66^\circ = 24^\circ$$ (Assuming $$\angle BAD = 90^\circ$$ which is not explicitly given, but the diagram suggests it might be a rectangle, or other properties are missing to solve directly).
• Let's assume there is missing information or properties for this specific problem to be solved directly with the given values. If ABCD is a cyclic quadrilateral, then $$\angle BCD + \angle BAD = 180^\circ$$ and $$\angle ADC + \angle ABC = 180^\circ$$.
• Angles subtended by arc CD: $$\angle CAD = \angle CBD = 42^\circ$$.
• Angles subtended by arc AD: $$\angle ABD = \angle ACD = 66^\circ$$.
• From these, we can calculate $$\angle ABC$$.
• $$\angle ABC = \angle ABD + \angle DBC = 66^\circ + 42^\circ = 108^\circ$$.

Answer: $$\angle ABC = 108^\circ$$

Problem 2 (Bottom Left): Tangent and Secant

• Given: A circle with center O, a tangent at point A, and a secant intersecting at points M and C. $$\angle B = 66^\circ$$.
• Find: $$\angle AMC = ?$$, $$\angle AOC = ?$$, $$\angle AKC = ?$$
• Analysis: This problem appears to have insufficient information or is incompletely drawn to solve. The angle $$\angle B = 66^\circ$$ is an external angle formed by secant BK and tangent AB. The relationship between $$\angle B$$ and inscribed angles or central angles is not directly provided without knowing other points or angles. Without more context or specific angle values related to the circle, these unknowns cannot be determined.

Problem 3 (Top Right): Circle with chords

• Given: Circle with center O, chord LK, $$\angle LPK = 42^\circ$$.
• Find: $$\angle LOK = ?$$
• Analysis: The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.
• $$\angle LOK = 2 \times \angle LPK = 2 \times 42^\circ = 84^\circ$$.

Answer: $$\angle LOK = 84^\circ$$

Problem 4 (Middle Right): Circle with chords and angle

• Given: Circle with center O, chord FK, $$\angle FEK = 98^\circ$$.
• Find: $$\angle FOK = ?$$
• Analysis: The angle subtended by arc FK at the circumference is $$\angle FEK = 98^\circ$$. However, this is an obtuse angle which is unusual for an inscribed angle subtending a minor arc. If $$\angle FEK$$ subtends the major arc FK, then the angle subtended by the minor arc FK at the circumference would be $$180^\circ - 98^\circ = 82^\circ$$. Or, if E is on the major arc, then the reflex $$\angle FOK = 2 \times 98^\circ$$.
• Let's assume $$\angle FEK = 98^\circ$$ is an angle subtended by the major arc FK, meaning point E is on the minor arc. Then the angle subtended by the minor arc FK at the circumference is $$\angle FCK$$ (where C is on the major arc) $$= 180^\circ - 98^\circ = 82^\circ$$.
• The central angle subtended by the minor arc FK is twice the inscribed angle subtended by the same arc.
• $$\angle FOK = 2 \times 82^\circ = 164^\circ$$.
• Alternatively, if $$\angle FEK$$ is indeed $$98^\circ$$, it might mean that E is on the minor arc, and it is subtending the major arc. In that case, the angle subtended by the minor arc at the circumference would be $$180^\circ - 98^\circ = 82^\circ$$. Then the central angle $$\angle FOK$$ (minor) $$= 2 imes 82^\circ = 164^\circ$$.
• If we assume the diagram is misleading and $$\angle FEK$$ is an angle subtended by the minor arc FK, then the angle at the center would be $$2 imes 98^\circ = 196^\circ$$ (reflex angle). The other angle would be $$360^\circ - 196^\circ = 164^\circ$$.
• Given the typical presentation of these problems, it's more likely that the angle refers to the minor arc. If E is on the major arc, then the angle subtended by minor arc FK is $$180 - 98 = 82$$. Then central angle is $$2*82 = 164$$. If E is on the minor arc, then it subtends the major arc. The angle subtended by the major arc is $$2*98 = 196$$. The angle for the minor arc is $$360 - 196 = 164$$.

Answer: $$\angle FOK = 164^\circ$$ (assuming it refers to the angle subtended by the minor arc FK)

Problem 5 (Bottom Right): Circle with chord and angle

• Given: Circle with center O, chord PT, $$\angle PFT = 60^\circ$$.
• Find: $$\angle POT = ?$$
• Analysis: The angle subtended by arc PT at the circumference is $$\angle PFT = 60^\circ$$.
• The angle subtended by the same arc at the center is twice the angle at the circumference.
• $$\angle POT = 2 \times \angle PFT = 2 imes 60^\circ = 120^\circ$$.

Answer: $$\angle POT = 120^\circ$$

Problem 6 (Bottom Middle): Circle with chords

• Given: Circle with center O, chords CD and PD. $$\angle CAD = 47^\circ$$.
• Find: $$\angle CBD = ?$$, $$\angle ADB = ?$$, $$\angle AOB = ?$$
• Analysis:
• Angles subtended by the same arc are equal.
• Arc CD subtends $$\angle CAD$$ and $$\angle CBD$$. Therefore, $$\angle CBD = \angle CAD = 47^\circ$$.
• Arc AD subtends $$\angle ABD$$ (from problem 1, assume it is given) and $$\angle ACD$$. We don't have $$\angle ABD$$ or $$\angle ACD$$ directly.
• Arc AC subtends $$\angle ABC$$ and $$\angle ADC$$. We need more information to find $$\angle ADB$$.
• For $$\angle AOB$$, it is the central angle subtending arc AB. We need an inscribed angle subtending arc AB, or other angles related to the center.
• Assuming the question intended to ask for angles based on the visible information:
• To find $$\angle CBD$$: Arc CD subtends $$\angle CAD$$ and $$\angle CBD$$. So $$\angle CBD = \angle CAD = 47^\circ$$.
• To find $$\angle ADB$$: We cannot determine $$\angle ADB$$ from the given information.
• To find $$\angle AOB$$: We cannot determine $$\angle AOB$$ from the given information.

Answer: $$\angle CBD = 47^\circ$$. $$\angle ADB$$ and $$\angle AOB$$ cannot be determined from the given information.

Header Text: 1. Найдите углы

This text translates to "1. Find the angles", indicating that the problems on the board are part of a set of exercises to find unknown angles.