To solve for all the angles in the given diagram, we need to use some key geometric principles such as vertical angles, linear pairs, and corresponding angles formed by parallel lines and a transversal.

Step-by-Step Breakdown:

1. Analyze the Known Angles:

• Angle 44° is given in the diagram at the top.
• Angle 78° is also given to the left of the point where multiple lines meet.
• Angle 64° is labeled near the bottom.

We'll use these known angles to find the other unknown angles: $a$, $b$, $c$, $d$, $e$, and $f$.

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2. Finding Angle $f$ (Using a Linear Pair):

• Angles $78^\circ$ and $f$ are on a straight line, meaning they form a linear pair.
• The sum of angles on a straight line is always $180^\circ$. $f = 180^\circ - 78^\circ = 102^\circ$ Thus, $f = 102^\circ$.

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3. Finding Angle $e$ (Right Angle):

• Angle $e$ is given as a right angle in the diagram. $e = 90^\circ$ This is clearly indicated by the small square box inside angle $e$.

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4. Finding Angle $c$ (Using Vertical Angles):

• Angle $c$ and the angle of $64^\circ$ directly opposite it form vertical angles, and vertical angles are always equal. $c = 64^\circ$

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5. Finding Angle $d$ (Using a Linear Pair):

• Angles $d$ and $c$ are also on a straight line, forming a linear pair, so they must add up to $180^\circ$. $d = 180^\circ - 64^\circ = 116^\circ$ Thus, $d = 116^\circ$.

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6. Finding Angle $a$ (Using Corresponding Angles):

• Angle $a$ corresponds to angle $44^\circ$ on parallel lines cut by a transversal. Corresponding angles are congruent, meaning they have the same measure. $a = 44^\circ$

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7. Finding Angle $b$ (Using a Linear Pair):

• Angles $a$ and $b$ are on a straight line, forming a linear pair, so they must add up to $180^\circ$. $b = 180^\circ - 44^\circ = 136^\circ$ Thus, $b = 136^\circ$.

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Final Answers:

• $f = 102^\circ$
• $e = 90^\circ$
• $c = 64^\circ$
• $d = 116^\circ$
• $a = 44^\circ$
• $b = 136^\circ$

Would you like any further details on these concepts?

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Related Questions:

1. What is the geometric rule for angles on a straight line?
2. How do you identify corresponding angles on parallel lines?
3. What is the definition of vertical angles?
4. How does the sum of angles in a triangle relate to these problems?
5. Can you solve for missing angles without knowing one of the angles?

Tip:

In geometry, always look for relationships like linear pairs and vertical angles to easily find unknown angles!